My problem most times is that because I check most of the character collision for the whole thing, if you go into a wall enough on both axes, it just puts you all the way out of an entire chink of tiles, as updating one axis the right amount won’t fix the other axis. I often use the old position on the opposite axis when calculating the first axis (though I think I just get lucky some times)
(Also, I wrote all this assuming we’re both talking about collisions where you push the character to the position they should be instead of reverting to the old position [also the vid doesn’t have a velocity more than 1 in each axis I think, so it doesn’t have to deal with that])
Edit: I don't think I made much sense, but to hopefully to make this clearer: the problems I have are mostly to do with going into walls with velocities that make it hard to tell exactly where the character has to be moved, as both axes have to move the character out the perfect number of pixels to make it look right, and you you don't hover or partially in the block.
The reason I didn’t spend much time on the character collision in this one because I just set the x to the old position instead of pushing you up against a wall (meaning you actually don't slide up all the way against a wall in most cases) ;)