Obed,
Ya done good! A near-perfect fall on your first try.
I've taken trees down both ways. As most of my felling has been for lumber and firewood, I usually use a saw. On the other hand, if the stump needs to come out, I prefer the backhoe. You should have no trouble with the size of tree you're working up to. My last backhoe was mounted on an 8N and had no trouble with trees up to 28" across. (biggest I ever tried, don't know where the limit is...) I'm by no means certain, but your hoe looks to be about the same size. Just going slow & careful means a small tractor can do many big jobs.
I keep seeing warnings & panic about dead limbs & broken trunks creep into this thread. These are problems whichever way you take the tree down. When I prepare to fell a tree using a saw, I inspect the whole tree before I start cutting. I do the same when I use a backhoe. It's called common sense.
Eddie,
One benefit I've always found to working from both sides of the tree instead of your preferred (dig both trenches from one place) method: I can dig like a madman from the down side & undercut the tree quite a bit safely. Then, when I dig from the other side, I can often watch the tree slowly fall away from me without pushing it at all. Kinda neat to watch...
(note...I started writing this immediately after looking at the last picture. I'm too lazy to go back & completely reword it if I'm duplicating anything someone else said before I finished...

)
Obed,
I now see you've already learned the "slow & careful" lesson well...
Only cutting roots on the "push" side of the tree is actually more hazardous than you think. If the "fall" side of the tree, for some unknown reason, has weak roots (it happens), then the tree can sway back & squash you when you go to push it over.

With a deep hole on the "fall" side, the tree will WANT to go where you're sending it.
Again, inspect the whole tree before you start. Hint: Lots of dead branches on one side of a tree often means the roots on that side have been damaged. (could also be a sign of trauma in the branches, but I wouldn't make that bet...) Of course, lots of dead branches would send me looking for a cherry picker to take the tree down in pieces.