They all are correct, depends on the maker, the region, time period, style of belt.
I make all three styles myself, two different woven styles and sewn loops. Depends on what the customer wants, look, style of the belt itself, etc.
I would have to say I've seen more original sewn loops on 1880 and later belts but those are also almost always a money belt which very few Cowboy action shooters wear.
A huge percentage of cowboy action shooters are wearing a belt that would have never been seen prior to the 1920's so the style of bullet loops being correct is pretty much a moot point for most.
Even in NCOWS many of the belts shooters are wearing are not very correct in one way or the other.
I guess what I'm saying is if your worried about period authenticity of the bullet loops you want to pay close attention to the belt itself and what style would have been seen on that belt for the period and region your rig is to represent.