Background: I've been riding a 2 wheel high-racer for the last month, have 184 km on it now, and have intimate knowledge of what you're going through.
You're a practiced Diamond Frame (DF) rider who can balance fine. To be brief, that confidence could be leading you astray. Why?
DF bikes are different. When balancing a DF bike you put weight on the pedals which takes weight off the saddle. Subconsciously you are "steering with your butt" by moving body weight left/right and back and forth. By comparison, a recumbent can only be steered by your hands, and to a small extent by your head and knee positions. Your bodyweight is completely static.
DF bikes have higher Center of Mass (CoM), but lower pedals. Recumbents have lower CoM but higher pedals. This means that it takes longer to get a foot down if things are going badly on a 'bent.
Recumbents are lower, so you're harder to see. My bike puts my face at eye level with a regular sedan driver, but on DF bikes my thighs/knees are about their eye level and I can see over things much better.
Length - Probably not an issue but most 'bents are a good 20-50 cm longer than a DF bike. This generally doesn't cause problems, other than slow speed manoeuvering or parking.
Steering - Most 'bents have steering in one of three formats. (This section is too long, look further down.)
Trackstanding/slow speed. As a beginner you simply can't balance a 'bent with no forward speed. There's a minimum forward speed you need to achieve to get balancing.
Pedals - some 'bent riders are happy with flat pedals. Some use the standard range of clips/cleats/toe cages, and there's also an option called a heel sling. To start with, ride flat pedals or put a spacer in the cleats, or flip them over. You don't need foot retention at the start. Later is a different question.
How to start
When newbies (mostly kids but beginner adults too) start riding two-wheel bikes, the current best practice is a Balance Bike. Since you're essentially starting from the beginning, balance bike is ideal.
Sit upright on the seat (aside, it's a "seat" on a bent and a "saddle" on a DF bike) without leaning back. Put your feet on the ground, and take alternate strides to move forward. This is scootering or scootching. Your aim here is to get up to at least 5-10 km/h and coast about. Don't try to pedal or lie back on seat. Your feet are near the ground to help catch any mistakes before they turn into a complete fall.
Your aim here is to get comfortable with the steering, and the loss of body/butt steering/balancing.
Try to do turns as well. Mind the front wheel - as it turns you might touch your legs. When riding normally it may be possible to have heel strike.
Once you're scooting around okay, lie back on the seat. Get comfortable scooting around, balancing and steering while laying back.
(If your bike has seat adjust, set the seatback to as upright as possible while you're learning.)
Once rolling, lift your feet up toward the pedals. Don't try pedalling, just get used to having them up high. Again, mind the wheel. If you get the feet onto the pedals, good work.
Get the feet onto the pedals as above, and then pedal. The pedal shaft should be somewhere between the ball of your foot and the middle of the arch. This is further backward under your foot than a DF bike, and this could be tripping you up.
Pedal about your quiet carpark and see how you go.
Avoid getting cocky or overconfidence at this point. You're still a noob (as am I).
Practice stopping and starting, without falling over. Starting from still is having the crank almost level but the rearward pedal slightly higher. Use your preferred starting leg, same as the DF bike. Ideally have the bike in a lowish gear on the big chainring for flat starts. Push the pedal and steer, while you bring up the other leg from the ground. It's got a long way to move compared to a DF bike.
Stopping is more a coast - you stop pedalling and brake. As your speed drops, put your primary foot or both feet down so they dangle but not touching the ground. Then gently brake to a stop and take your balance.
At first your track is going to be wide - you'll need 2+ metres to ride about around with reduced risk of tapping into things.
At speeds below 5 km/h, especially while starting off, you need some space. Don't get yourself into a position between cars where there is insufficient space. Take the lane at traffic lights rather than filtering.
You're going to fall while learning. That's a given, even more likely than falling with cleats on a road bike.
Here's my first fail - I was coming to a roundabout and didn't want to stop so I kept my feet up. But I slowed down too much and overbalanced at about 1 km/h. No damage to me but I scared the driver of the black car.
Second fall was not captured on video. I was trying to turn from a driveway onto the road. Again, slow speed plus the uphill slope of the road camber and I was too slow to balance.
Common thread here is "Too slow and I fell"
If you're on strava, feel free to explore https://www.strava.com/activities/1617908904
which is my first hill ride. Around 33 minutes in I hit the first grade which is about 10-11% at the start, and then relaxes to 8-9%.
My speed was 4-6 km/h in bottom gear which was very wobbly - I was easily taking the whole car lane to keep moving up the hill, and I still had to quick stop a couple of times.
Minimum safe speed on a two-wheeled 'bent is 5 km/h on the flat Any less and you lack steerage-way.
How to fall
If you're going slow enough you can stick the downside foot out and try to slow down the fall.
Risk here is if you're going too fast (above ~4 km/h) that your foot will touch the road and then be thrown backward under the saddle, (or under the frame on a trike!) This is called leg-suck or legsuck and can result in bruising, broken bones, and a serious loss of control.
Another problem with leg down once the fall has started is that the foot is quite a long way forward and your CoM is further back. So one counter-intuitive assist is to sit up and bring your torso closer to the bars.
DO NOT ATTEMPT TO CATCH YOURSELF WITH YOUR ARMS/HANDS The angles are all wrong, and even a stopped sideways tilt will hurt your shoulder. I've already done this and its been aching for about 10 days now. You can also fracture bones in your wrist trying to take the force.
Best thing to do is have your feet down early ready to act as landing gear. Bents with wider seats might just be better to let drop from under you. Narrower seats will make your hip hit the ground first, which would be bad too.
All DF bikes have a forward-facing stem and then a sideways bar, which is such a common system it has no name. The bar's grip describes a circle of motion that matches the rotation of the front wheel. When applied to a bent it might be called a 'tweener bar (because the rider's knees go between the bars and the stem) and looks like this:
OSS/ASS Overseat Steering / Above Seat Steering (yes what a bad choice of acronym) This is where the stem faces backward from the steerer. The grips describe a curve that matches the direction, but its in the opposite direction. This is like an outboard motor or an early car.
USS (underseat) direct, (more common on trikes) where the grips are by your hips or thighs, and have a pronounced left-right movement to turn. These are connected by a stem directly to the steerer tube of the fork, but that stem goes aft and then the rider steers like tiller steering, except its below you. Not an ideal image sorry:
USS indirect - Similar to above. There are grips on either side of your hips, that move fore/aft and do not share a central pivot with the steerer. Has a pushrod for a two wheeler, or ackerman steering for a trike. Looks somewhat like this - the pushrod is the black line under the boom and bottom bracket/crank axle:
Finally there are oddball systems like this, which have a Universal Joint at the top of the steerer to give the rider a "steering wheel" like a car. These are uncommon.
Some OSS systems have a hinge to allow the handlebar to move up or down.