Alright, after watching your videos and sitting down in front of a proper keyboard, here are my thoughts (edit: this turned into a gigantic wall of text, so I put a tl;dr at the bottom. I do encourage you to read the whole thing, though):
FIRST AND FOREMOST, I don't condone training at home without a coach. Having said that, I know telling a gymnast not to tumble is like telling a cat not to chase mice, and if you're gonna do it anyway you may as well do it with correct technique. So I'll give my technical thoughts below, but let the record show I do not condone training flipping skills at home.
The backhandspring is not too bad; when you described it as undercut, I was picturing something much worse. Having said that, here's my favorite backhandspring progression, which is always good to come back to at any level to reinforce solid back tumbling technique:
1) Standing backhandspring without a full arm swing. Stand with your arms horizontal or slightly below horizontal in front of you, your knees bent, and your back rounded. Sit back, and do the longest backhandspring you can. The longer the better (within reason -- still focus on tight legs and other aspects of form). Hands angled inward the whole time, especially when they contact the ground. Stick with no rebound; your stick position should mimic your starting position as closely as possible, with arms in front, knees bent, and back round.
2) Standing backhandspring as described above, stick with no rebound, then do a second backhandspring from that same spot. You can (and should) stop as long as you need to, but do not move your feet or swing your arms between the first and second backhandspring. Again, you want these backhandsprings to be as long as reasonably possible.
3) Same as above, but this time without coming to a complete stop between the first and second backhandspring. The connection should be more or less in slow-mo.
4) If you have enough space, do the same as above with sets of 3 or 4 backhandsprings instead of just 2.
5) Let the pause between backhandsprings evaporate, leaving you with long, stretched out, connected backhandsprings with smooth transitions. Don't try to go fast; try to go long. You'll find that by aiming for distance, you'll "accidentally" find yourself tumbling faster as well.
I discussed these progressions in a clinic I did at woodward a few years back, which one of the other coaches uploaded to youtube:
This also has some drills you might do for your roundoff as well. And remember: it's always worth coming back to these basic drills repeatedly at any level. I used to use almost this whole progression as a warm-up for my athletes.
Now on to the back tuck.
Before we even get to the specific drills and progressions, here's something I want you to focus on in every single flipping skill you ever do for the rest of your life: pay attention to what you see while you're doing it. Not just where your head is facing, but actively try to notice what you see while you're in the air. Try to see and notice details.
If you don't know what you see, think about keeping your eyes open and just try to notice what you see.
When you just see a blur, try to notice the differently-colored blur depending on whether you're looking at the sky or the ground.
When you notice the color differences between the ground and the sky while flipping, try to notice details. Try to get a deliberate and detailed look at the spot on the ground where you're going to land.
This takes time to train your brain and eyes to work together like this while flipping, but it will make a huge difference in confidence, consistency, and control (and possibly even benefits that don't alliterate).
The biggest challenge of a back tuck is retraining your natural intuition about rotating. Almost everything that your brain uses to measure balance and rotation is in your head; because of this, your subconscious intuition is that making your head rotate makes your body rotate. This is not specific to you; this is how the human brain works.
But this intuition is dead wrong, and that's why you (along everybody who's ever learned a back tuck) instinctively throw your head back when you want to rotate backwards.
Two things to focus on while doing a back tuck in order of importance (and these are all considerably easier to fix on a trampoline than on harder ground):
1) Lift the center of your chest as you take off. Extend your hips and knees as you take off to make sure your chest is rising as aggressively possible on takeoff. This will reinforce proper takeoff position, particularly through the hips
2) Pick something in front of you, and try to watch it until your knees block your view. In the second half of the flip, it's not a problem if your head goes back to look for the landing, just make sure you see something in front of you and then see your knees first, then look for the landing. This will reinforce proper head position and use of visual cues.
3) Turn your hands so that your palms face outward. Focus on keeping your hands turned this way while reaching your pinky fingers towards a spot on the ceiling (or in the sky) above and just slightly behind you. This is to reinforce good arm and shoulder extension on takeoff.
Again, I'm not giving you this technical breakdown because I think flipping at home is a good idea, I'm giving it because if you're going to do it anyway you might as well do it correctly.
1) I don't encourage training at home without a coach, but if you're gonna do it anyway....
2) Work on standing consecutive backhandsprings, going for length
3) Focus on what you are seeing as you do the back tuck, and look for something in front of you during takeoff
4) Try to lift your chest upward as you take off the back tuck