This answer draws some points from my answer on the commuting question that the OP linked to.
There are two characteristics that we may be conflating. Lenses have a certain base color, which affects what you see when you look through the glasses. To my knowledge, these include at least grey, bronze (aka brown), rose (aka pink), and yellow. These are the common ones I see in cycling.
Then there's the lens' mirror finish, which affects the color that others see when they look at you. I don't work in the optical industry, so I don't know this for a fact, but it seems like the finish may not significantly affect what the wearer perceives. It's definitely possible to make base colors in more than one finish. For example, Oakley offers bronze, gold, and reddish lenses; all of these have bronze base colors, and the latter two also have a mirror finish. The visual light transmission ratings (see below) do differ slightly among these lenses.
Separate from all this, visual light transmission (VLT) is the proportion of all visible light that the lenses allow to pass through. I am most familiar with Oakley glasses, and to my knowledge, most of their lineup has 10%-20% VLT. For cycling in the peak of the summer, some cyclists may perceive lenses near 20% VLT as inadequate. This depends on your preference. I'm not sure if some mirror finishes are associated with lower VLTs.
The original question asked,
Would it be adequate to stick to neutral-colored filters (which would presumably be ideal for enjoying the scenery), or do any particular colors provide a significant advantage?
I believe this answer deals mainly with the base color of the lens. This probably involves some element of personal preference. Grey base colors reduce the intensity of incoming light evenly across all wavelengths. Said another way, they don't alter your color perception. Bronze and rose base lenses both increase contrast, and they may alter your color perception a bit (with rose lenses possibly altering your color perception more than bronze, based on my personal experience only).
What's contrast? There are technical definitions of the term, which I will skip. In cycling, increased contrast can help us to perceive obstacles like potholes on the road or rocks on the trail. As an extreme example, consider the two photographs of the surface of Titan, one of Saturn's moons. The images are taken from the Wikipedia entry. The left image has no contrast enhancement. The right image has been artificially contrast enhanced. Obviously, we aren't going to cycle on Titan's surface anytime soon, so the images just illustrate the principle.
To answer the question, I think that neutral base lenses are definitely adequate for cycling. It's probably worth considering lenses with bronze or rose base colors at some point. Again, this is likely to be an issue of preference. Oakley's Prizm Road and Prizm Trail lenses, which are intended to be specific for road cycling and MTB respectively, have rose base colors. Smith appears to offer cycling lenses in both grey and bronze bases. (NB: these aren't product recommendations, these are two major manufacturers I'm familiar with, Oakley more than Smith). My own experience is that I like lenses with contrast, so I tend more towards rose and bronze bases for cycling.
There are other base colors, e.g. green bases appear to be marketed for golf, yellow/amber bases are marketed for low light conditions, blue bases may be marketed at snow and water sports. I'm not as familiar with these. They don't seem to be commonly marketed in cycling.
On the broader topic of color enhancement, some manufacturers are further tuning which colors are enhanced with the aim of enhancing visual perception. That is, they use dyes to block more light at certain wavelengths. Oakley's Prizm and Smith's Chromapop technologies use this principle, and other manufacturers may do this also. With Oakley, some of their Prizm lenses are marketed as being specific to certain sports. For example, the road lenses are said to enhance yellows, greens, and reds - all of which correspond to standard colors for traffic lights and road markings. They also have general purpose Prizm lenses (branded as Prizm Everyday), which are said to just make most colors more vibrant. I have a pair of Oakley's Prizm Road and one of their Prizm Everyday lenses, and to be honest, I think they are nice, but I'm not sure I could differentiate the effect from placebo, and this type of technology is definitely not essential. Naturally, feel free to experiment or not.
Do photochromic (a.k.a. photochromatic or Transitions) lenses vary the darkness ideally, or is it really better to have multi-lens glasses with a few different colors?
To my knowledge, all the current photochromic lenses offered for cycling have grey base lenses. Based on personal experience and listening to cycling forum conversations, I'd assert this: photochromic lenses require UV light to darken. They don't seem to get to their darkest tint unless it's very sunny. At the time I tried them (several years ago), their maximum darkness (i.e. minimum VLT) didn't feel dark enough for the sunniest days. I'm not sure how much the technology has improved in the last few years. They might have a niche in cycling for riders who are competing in extremely long events that start in the morning and finish at night (or you could just remove your sunglasses when you don't need them). Other than that, I don't see that there's a strong rationale for transitions lenses.