I've always viewed these differences as matters of random fortune or misfortune.
They are, in fact, something you can create with exact specificity, as if you had a dial... which a maker does in yogurt by controlling milk composition, make/treatment, and culture selection.
but are there ways to achieve noticeably different styles with just cultures and process?
Yes, anything from runny to stand-your-spoon-in thick can be accomplished with bacteria and temp alone. Process does help in the sense that scalding denatures whey protein and caseins adsorb it, making for a thicker set.
but I never really noticed any difference in the yogurt I made with any of the different brands.
By and large, those are all standard formulations. If you are after specific textural attributes, you have to be very strain-specific in your choices. Especially when it comes to acidophilus and Strep salivarius thermophilus. The major culture houses have specific strains you can use to achieve the taste you want (mild to tart) and the texture you want. Yogurt texture through bacteria is achieved by the various exo-polysaccharide structures (EPS) present on the cell wall. These enable cells to form long chains, and give body and texture to the lactic gel. Anything from ropy and thin to thick.