lazy and theif vaults, unlike most others, are done when you're not running at the obstacle head on (for example, running parallel to a rail, you'd lazy to get over, while running at it perpendicularly [head on], you could kong, side vault, dash, etc.)
So back to running parallel to a rail, if you're running beside the rail, lazy over, end up facing the same way, and keep running parallel, it's a lazy vault.
Lets say you're running and you want to go the other way; a thief vault is essentially a lazy, but you end up facing the other direction. After the vault you are now both on the other side of the obstacle, and also running the other way.
or such is my understanding
EDIT: also, what BobT said.
Bionicgrape, too, is partially correct, only the vault isn't done immediately after you steal something, but only after you've managed to outrun at least 85% of the policemen/security officers chasing you. (some freerunners will tell you you must lose only 65% of your pursuers. These freerunners are wrong.)
Run till you don't here sirens, that's my policy. Or run to hide, find an alley about your width and Spiderman Climb between them up and out of sight. lol.
But JFlash is correct, a thief changes direction, which is a very VERY good technique to learn.
I get it but why make it confusing and give it two different names if its just the landing that changes. Yea I would get it if you stole something and decided to do the move and call it a thief vault lol..... just lazy vault sounds two boring.
I tried doing it with the opposite foot and it just feels wierd and not right and at the end it still looks the same its just harder with the outside foot
killian's correct in his reverberation of bobt's post. Go outside and do them and you'll notice theifs use the other foot to kick around, so as to swing it and change direction. It's kinda like a speed