I'd like to expand on the answers offered by @STW and @Mac.
The current rules here ask us not to offer product recommendations (the rules at the time of the original question may have differed). I'm going to focus on what to look for instead of specific products.
I believe the OP had glasses that weren't sport-specific, which Oakley may call lifestyle glasses. I personally prefer cycling-specific or wrap around glasses. The objective difference is how much each type of glasses cover your field of vision. The wrap around glasses offer more coverage, including in your peripheral field of vision. This means it's less likely for dirt to get blown into your eyes.
Another possible issue is that sunglasses frames can get into your field of vision. Frameless glasses will avoid this. If you are a more performance-oriented cyclist, then sports-specific glasses often position the frames so that they aren't in your way when you're in a riding position. Lifestyle glasses are presumably designed for vision when you're not hunched over, so their frames may get in the way if you're on a more performance-oriented bike. However, if you have your commuter set for an upright position, this may be moot. In any case, it may also not be a major issue on a short commute.
I don't think it is necessary for commuters to get sports-specific glasses. If you have lifestyle sunglasses and you don't feel like you're getting wind or dirt in your eyes, then there's no barrier to just re-using them.
A subjective disadvantage to sports-specific glasses is that you may perceived as unfashionable or odd if you wear them off the bike. But this is subjective.
More serious cyclists will ride longer distances, and they may ride in more adverse conditions. I think more serious cyclists should consider sports-specific eyewear. As a tangential addition, sports-specific eyewear does not require additional retention systems, relying instead on special purpose rubber. In my experience, this is more than sufficient. General purpose or lifestyle sunglasses may not have as strong retention, and users might want to think about a strap if they find their glasses slide off.
How dark are they (visible light transmission)
Sunglass lenses range in how dark they are. Often, this will be summarized by the visible light transmission (VLT) figure. I recall that the darkest Oakley glasses (the only ones I'm familiar with) offer VLTs of 9-12%, i.e. 9-12% of visible light hitting the lens gets let through to your eye. This amount of VLT is fine for very bright light. In cloudy conditions, some riders may perceive those lenses to be too dark. Alternatively, some riders may be more sensitive to light. Generally, a manufacturer's product webpage will tell you the VLT of a lens. A retailer should be equipped with this information as well. It's a key piece of information.
Selecting VLT is a matter of personal preference. I have been riding a pair of Oakleys with a VLT of 20%. This does initially seem quite high, but I found that it has actually been sufficient on sunny days. I'm familiar only with Oakley's offerings, and I recall that most of their lenses have VLTs ranging from 10 to 20%. I have a feeling that many people would find VLTs of 15 to 20% acceptable for general use, i.e. they'll be usable on very sunny or fairly cloudy days. However, once again, you should let preference be your guide and you may have to experiment a bit. As noted in some answers, some glasses allow you to interchange lenses, so it's possible you could get multiple lenses with varying VLTs. A side note: some manufacturers may make clear lenses with 75% or higher VLT available for cloudy days or dawn or dusk riding.
One answer mentioned transitions lenses (aka photochromic lenses). These change the amount of VLT when exposed to ultraviolet light, and they can be useful for changing conditions (e.g. you start a ride around dawn, and you ride back in full sunlight). Oakley's literature quotes a VLT figure of 23% for their transitions lenses, which is relatively high. Do consider this. Also, my experience with transitions lenses is that they don't achieve their full darkness (i.e. minimum VLT) unless you're in very bright sunlight. However, transitions technology may have improved since I last used them (around 2015 or so). Transitions lenses are generally available in grey base tints (discussed below), and I'm not aware of transitions lenses with other base tints.
Contrast and tints
Lenses also have a base tint, i.e. what you see when you look through the lens. This is not the color of the lenses when you look at them from the outside (i.e. as perceived by others); that appears to be a mainly cosmetic thing.
Some lenses have a gray base tint, i.e. they look gray when you look through the lens. That's considered to be a neutral tint. It affects all colors equally, and it doesn't affect your color perception.
Some lenses have bronze or rose tints. Both those types of tints increase contrast, i.e. the contrast between light and dark. Borrowing a graphic from the site All About Vision:
Bronze (synonyms may include copper, orange, or amber) or rose base tints may help you see the last row of letters better. In addition, blue light may affect your eye's ability to see details or to focus properly, and bronze and rose tints block more blue light than grey tints, as discussed at Sailing World. This can help you distinguish potholes or other hazards on the road, but if you are only commuting, this may not be a material benefit. This Wirecutter article contains a quote from Rob Tavakoli, who runs the online glasses store SportRx. He endorses contrasting tints for outdoor activities, explaining that they can really enhance your vision when traveling at speed. However, the site does say that more leisure-oriented riders or commuters should be fine with a neutral tint.
I'm less familiar with yellow tints. They should give some of the contrast benefits discussed above. In my limited experience, they tend to be mainly designed for low light conditions (i.e. pretty high VLT, perhaps 60% and over). Cloudy days would be an example. I don't see green tints offered for cycling eyewear, but I believe they may offer a more neutral color perception and may be favored in some ball sports.
Some people might find some tints to be off-putting. Rose tints do alter your color perception, which may be a further turn-off. Again, you may need to experiment and consider your preferences.
If you can't decide, the safest choice should be a neutral grey base. Personally, as a performance-oriented cyclist, I find that I like rose or bronze tints due to the added contrast.
Polarized vs not
Polarized lenses block glare from sun reflecting off car windscreens. Many driving glasses are polarized, since there can be a lot of glare when driving on the highway in sunny conditions. Ice or snow on the ground also causes glare.
It seems like many cyclists see that polarized glasses are more expensive, so they get them. I differ from this opinion. Many of us may not be in heavy automobile traffic, so glare reduction may not benefit us. Cycling on sunny snow days would seem like a use case where polarized might help, but how many of us are actually cycling in those conditions? In addition, we may actually benefit from seeing glare reflected off a puddle - it may help us perceive the puddle earlier. Further, while this may not be a consideration for many, polarized glasses can make your bike computer screen or smartphone screen look odd, and may impair readability. Last, polarized glasses may slightly affect your depth perception, as discussed at the Wirecutter article in the last link.
I would assert that polarized glasses are not necessary for most of us. Getting polarized lenses should be a matter of preference. If you're commuting in with your driving sunglasses, polarized lenses won't really impair you - my issue is with the contention that they are necessary for cycling.
The argument above is, admittedly, taken from videos by Oakley that discuss their rationale for the Prizm lenses - their cycling (and ball sports) Prizm lenses are not polarized, although their water and snow sports ones are. I have no commercial relationship with Oakley to disclose. I am merely a frequent customer.
Side note: altering color perception
This is likely not that important to the OP's question, but independent of the tint, manufacturers may be able to block more or less light on certain light frequencies using dyes. The rationale here is to emphasize colors that are important in specific environments. This is independent of the base lens tint. Oakley's Prizm and Smith's Chromapop lenses are examples of this type of lens technology. Oakley offers Prizm lenses customized to various sports, e.g. road cycling, mountain biking, shallow water sports, deep water sports, etc. Each lens highlights colors that are critical to perception in its sport. Oakley also seems to offer a line of general-purpose Prizm lenses, and Smith's technology also appears to be more general-purpose. Other manufacturers might use this technology, but I'm not aware of them.
This is definitely not vital for most consumers. As a more performance-oriented cyclist, I do like Oakley's Prizm Road lens for road cycling. It has a rose base tint, plus the lens enhances the transmission of yellows and a few other colors (to emphasize lane lines in the road). It was designed specifically for road cycling. However, it does seem a bit odd when I drive with it. In gravel cycling, it definitely seems a bit bright when I'm on gravel (which often contains yellows).
If you get a lens with very sports-specific light absorption characteristics, be aware of the trade offs in other environments. For a commuter or leisure cyclist, I'd recommend holding off from a sports-specific lens unless you know you're committing to the sport. As I mentioned, general-purpose versions of this type of lens exist, which some might want to consider; in fact, I'm trying one out right now.