I grew up a "traditional" bowler, but I've toyed around with both two-handed and the no-thumb method you speak of. I mention this just to let you know that I've dabbled in many styles myself.
Okay, as for converting to two-handed, first of all, do YOU want to do that? If so, why? You said that some higher average bowlers around there told you to give it a try. What reason(s) did they give for that? I could see one advantage being that having that second hand on the ball can offer a bit more control/stability compared to the other no-thumb methods. Still, guys like Mike Miller, Tom Daugherty, and Patrick Girard (I think that's who I'm thinking of) have/had a lot of success with one-handed no-thumb bowling. Again, before deciding to switch, be sure to ask yourself why you're doing it.
Your two-handed shots actually showed one really nice thing; you are flexible enough to get your left arm across your chest in the backswing. That is something I always struggled with big time. That's important because if you can't do that, you can't generate speed, and your left hand is likely to come off the ball. Both of those things cause big problems!
Now, in regard to issues, here's what I saw:
1) To me, it looks like your left hand is off too far on the side of the ball during setup. I think it works best when the fingers on your left hand are basically touching end to end with the fingers on your right. So, basically, the left hand is pretty much on top of the ball. I think that's one of Svennson's big advantages. The kid is so lanky that he can pretty much keep both hands connected. Because of that, the ball is 100% secure all the way through until release.
2) I'd consider 5 steps instead of 4. I say that because two-handed bowling seems to go best with a certain cadence to your footwork. We found that it was best to go 1...2-3...4-5. The skip or shuffle step came between steps 3 and 4. I'm not saying you can't bowl two-handed with only four steps, but I always found the aforementioned rhythm very important.
3) On your down swing, you need to work on getting the ball down into the lane. Right now, both one handed and two, you like to let your hand slip out around the side of the ball very early. That's what causes your ball to kind of flop onto the lane instead of being projected onto it. I've always been told to think of it like landing a plane. You don't want to come in too steep and crash hard onto the runway, but you also don't want to flop up and then back down. The key is to get the ball into the lane.
4) One thing that may help you get the ball into the lane better is more knee bend. You don't want to throw the ball with a stiff/straight left leg. If you get down lower, it's going to be easier to bet the ball down into the lane properly.
5) Lastly, going along with No. 3, you need to work on keeping your hand under the ball longer and letting it roll onto the lane instead of dropping it there. Tom Daugherty really does this well. Even though he isn't using his thumb, he stays behind the ball nicely and seems to push the ball out onto the lane with his fingers rather than ripping up around it and flopping it out there.
Obviously, that's a lot to think about. Again, start by asking yourself what you're trying to accomplish and what pros and cons there are to sticking with what you've been doing versus going to two-handed. Once you have that worked out, practice, practice, practice. While doing so, be patient, and take one thing at a time. It takes a lot of time to ingrain proper muscle memory. In the end, no matter what happens, make sure you're having fun; that's why we bowl in the first place!