One distinction I see - I didn't notice it in the linked threads, but one of them consisted largely of more links, and I couldn't follow all of them - is that "team" has two meanings: the group of players who are on the field* (or could be) in a given game, and an organization. Thinking of it as an organization, one plays for the team. Thinking of it as players on the field in a given game, one plays on the team, at least in AE.
As Biffo posted, context is important. Here, it is essential.
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*Or court, pitch, ice, whatever.