Since you're paying attention, I'll elaborate a little. When left at room temperature, as I'm sure you know, dough ferments faster than when it's kept in the fridge. With faster fermentation (or faster rise), a laminated skin is likely to merge back into one layer. I'd say two hours at room temperature is way more than enough time for that to happen (but that surely depends on yeast percentage and hydration/oil percentages).
I know fazzari advises keeping the skin cold until right before baking, and he does it that way for the same reasons I do it that way, even though each of us came to the same conclusion totally independent of each other. As you've already figured out, fazzari's work is a very good model to follow with this style.
Also, just about any parbaked skin will end up with the appearance of lamination because the whole skin usually becomes a giant bubble while it's parbaking. (Is that what yours did?) As I said earlier, this creates a crust that kinda looks like lamination, but it's just not the same to me. But do it every way you can think of. That way you'll know from experience how everything works, rather than thinking you know something just because I (or anyone else) told you something.
One more thing: I like AP flour for this style. I've recently used some higher-protein flours to do laminated cracker (using fazzari's "Every trick" instructions) and had good results (at least when I used considerably higher hydration than when I use AP flour), but I think I like AP better. Ideally I'd probably like to find a foodservice-quality flour that's maybe a little higher protein than AP, but for now I like Pillsbury and Gold Medal AP flour for laminated cracker style.