On the subject of which of the questionable team members count... My opinion is that Larvitar, Beedrill and Haunter count, and that Seaking definitely does not. I'm happy enough to go along with the points that have been made, in agreement with my own opinion of course, for Larvitar.
The answer to the Haunter debate is simple however; he joined Ash's team of his own volition (as did Charmander, Squirtle, Heracross, and more) and left to stay at Sabrina's with Ash's blessing. The length of the tenure here doesn't matter so much as the conditions of it.
Beedrill was a legitimate capture and was a member of Ash's team if even for a very short amount of time.
Seaking was not caught to be kept, but was merely the means by which a fishing contest was decided.
If anybody wants to disagree, then feel free, but try not to run me down if you do. It's only MY opinion after all.