Thank you for pulling it down off the shelf and adding your observations. The photos do reveal some kind of texture on the "smooth cover" copy, but in person, it feels substantial different. I'd think the cover with the raised texture would wear on the upper surfaces and create a kind of denim / canvas aged look. Agree about the bleed-through copy inheriting some aspects of earlier prints.
Here's a good current example of a 2nd print from eBay:
** expired/removed eBay auction **It has two commonalities with the first copy: distinct raised texture on the cover (noting the wear), and the title page going further into the gutter. The copy with the smoother cover has the title page far more centered title page - I'd imagine your copy fits this as well.
eBay (2nd Print):


Two copies on hand compared:


So perhaps Third Alpha has some variations in the run, whether cover stock or page weight or both. If any collectors have enjoyed the discourse so far, would be excellent to compare against a Second Print and Fourth Print (or Third Beta) on hand.