If it's literally only a few seconds, I'd count it as one set, but probably still strive for being able to do them without the rest.
However if you begin pushing it up to 10 seconds or more, then you're getting in to rest-pause training, which is a technique where you effectively extend a set beyond failure by taking a very short rest, performing more reps, and repeating. Although it's usually seen more as a way to compress multiple sets into one, rather than getting more reps into a single set. Rest-pause training is more effective for hypertrophy than strength1, and is popular both for bodybuilding and just for saving time in the gym.
As for not being able to repeat the same number of reps on subsequent sets, this isn't surprising. Pull-ups are quite fatiguing and I think not being able to repeat a set performed to or close to failure is quite common. It's still training strength though, it's just that the systems responsible for performing strength work (the phosphagen energy system and possibly the central nervous system) have been diminished by the first set. If your subsequent sets are seriously diminished, then it may be worth considering backing off a little earlier on the first set so that you can get more high quality sets in. I.e. 4 sets of 5, rather than sets of 7, 4, 4, and 3.