HH are hand hammered and have wide (traditional) lathing similar to AA.
HHX are hand-guided machine-hammered, and may or may not also be hand hammered. HHX have tight lathing similar to AAX.
That's exactly right.
I was under the impression that the HHXs were just hit with a larger headed hammer, as the HH in HHX stands for hand hammered
The HH in HHX stands for "this is an offshoot of the Hand Hammered series that probably doesn't have any actual hand hammering on it". While the small hammer strikes on the HHXs are still done by hand, most HHX don't have those - most only have the jumbo hammer strikes which are done by hand-guided machine, as Anduin pointed out.
Just to repeat and perhaps emphasize the point - Zildjian does exactly
zero hand hammering on any of their cymbals. That includes every single model within the K, K Custom, and K Con lines. Go to the Zildjian website and look at the very first question on the FAQ section where they're very clear about this. However, and this is just my opinion, that doesn't make them any less of a cymbal. There are too many other factors that go into designing and making a cymbal that makes a greater contribution to the end product than just how those hammer marks got there.
Full disclosure: I have Zildjian As, A Customs, and Ks as well as Sabian AAXs and HHXs in my current collection, and have had AAs and HHs in the past. I love all these cymbals so to me it really doesn't matter, though it is kind of interesting
I'd wager the X is, of course, more modern sounding and a bit brighter.
Not to sound like a jerk ranking on Sabian (not my intent), but honestly, the "modern cymbal" claim is a dubious marketing construct carefully designed to get you interested, and should be taken with a grain of salt. It doesn't really matter what they call them since you should be checking them out and fully digging them before you buy anyway. But the way I see it, nobody can rightfully claim to churn out "modern" cymbals when they're using more old-world Turkish cymbals formulas and processes going for that darker smokier vibe reminiscent of the old Ks. The way I see it, Paiste are the masters and owners of the "modern cymbals" crown with their 2oo2s, Giant Beats, and Signature alloys. That's really quite a departure from what I think of as traditional Turkish cymbal-making and the more industrialized North American off-shoots.
Anyway, pound for pound, the HHXs seem to be actually darker to me, which probably makes sense since there's quite a bit more interruption of vibrational paths through a heavily jumbo hammered cymbal than a more lightly hammered, less disfigured cymbal. But yeah, a thicker HHX Stage Crash is going to be brighter than an HH Thin Crash, so you have to get as close to apples to apples as you can to make that comparison.