There was certainly an element of competition among lifters, as there is with any good program. If lifters aren't competitive, they won't thrive in sport. I've heard Abadjiev say that he often invited spectators to watch training to simulate meet conditions and force better performance, but that/competition aren't the elements that characterize what's considered the Bulgarian method - that's training max and near max singles very frequently. You'll never be able to define it perfectly because it was never exactly the same thing, as Abadjiev was constantly refining it. He's now coaching about 40 min from us, and a former lifter has come here full time. Their programs are basically front squat to heavy single, then 2x2 back off, snatch to heavy single, CJ to heavy single, then front squat to heavy single with 2x2 back off. However, when one was in more of a learning stage, he had her do pulls occasionally. Another sometimes does rack jerks. I know a former one from his first time here did overhead squats. He also plans intensity variation day to day - I've been shown a weekly programs that had numbers planned for each lift, modulated up and down day to day, which goes against the daily max idea in a sense, but really all he's doing is predicting what the daily max will be and giving goals.
I think it was Steve Gough who said with regard to the Bulgarian method, there is the letter of the law, and there is the spirit of the law. You can get a lot done with the latter if that's what you choose to do.