Ude hishigi juji gatame – the cross-body armlock – has two major configurations in judo. In the first, one leg is over the head and the other behind the back. This is the one that Catch refers to as the “Japanese armbar”. It’s the first one I learned as being the “traditional” way to do it in judo. The second configuration is one leg over the head and the other over the body. I’m pretty sure this is the default in BJJ.
The first configuration is generally more applicable in a context where you’ve just thrown uke, especially with an arm or hip throw that places them in front of you in an almost knee-on-belly position already. It’s also good from knee-on-belly for the same reason. It’s faster from here because you don’t need to get the leg over the body. However, if you’re doing this variation you want to be off to a 45 degree angle from their body instead of perpendicular to them. You should actually use the foot behind their back to push slightly and the one on their head should be biting in to make that angle as it lifts their shoulder up, reducing the amount of give and preventing them from being able to sit up into you as an escape. Pointing the thumb toward their hips (as opposed to straight up or towards their head) helps prevent a Hitchhiker escape and will tighten the lock. Some people like to hook into the armpit with the foot over the head. The big reason is that it prevents the escape where they just push the leg over their head. I’m not personally a huge fan of that and as Mike had noted the other night at class being on the head helps keep more pressure in the position. It’s nice for a transition to reverse sankaku though.
The second configuration is better from mount or closed guard because you don’t need to clear the leg to the other side of the body, and in my experience, is probably the better control position. The Spider Web from here is often easier for me, but it’s also a nice position to set up a bicep crush or jigoku jime (which has apparently been renamed to “Hebrew Necktie” by 10th Planet when done for no gi? This is an old technique, the no gi variation was up on LapelChoke back when that was around). I’ve always been taught for this configuration to point the thumb straight up to the ceiling. Switching to sankaku here, I’ll thread the leg that’s over the body through the arms and over the shoulder. Almost everyone sits up thinking that I’ve given them the escape, but really I just had a hard time breaking their grips and decided a triangle would be easier. You can switch the legs (threading the leg near their head through their arms) for the same reverse sankaku you got from the first configuration, but again, it’s more work than threading the body leg.
So, the big differences? When you use it (KoB vs mount), what angle you want (45 vs 90), and the hand rotation (toward belt vs straight up). Everything else is situational so it’s not as clear cut. You can always throw your leg over the body one way or the other to swap between them as you need too.