The difference, as so often, is scale. If you poll 500 people in a congressional district and ask them what their reaction to Candidate X's drinking problem is, that's an information gathering poll, not one intended to sway a significant number of voters. It might not be "scientific", but it is intended to gather information about broader voter sentiment to guide your campaign strategy lines, because even changing 500 voters minds isn't going to have much effect on the race. If you did asked the same questions, and even made the same decisions based on the data, but did it to 100,000 voters then it does start to look more like a hybrid or closer to the push-poll side of things.
I disagree strongly that scale has anything to do with it whether you call a poll a push poll. Scale is irrelevant. Asking what you think about a candidate's drinking problem is a push intended to communicate the idea that the candidate has a drinking problem, without regard to context and without discussion. You're right that it's not scientific. You're wrong that it's neutral information gathering. It doesn't matter how many people you ask that question. You can ask the same question to a single person and it's still a push, or more commonly called, gossip or shade.
Your mistake here is assuming that a push poll needs to be influencing a "significant" number of voters. It does not. Again, the defining characteristic of a push poll is the push. Not the number of people surveyed, not whether the pollers are collecting results, not whether they're looking for an angle of attack versus attacking. If the poll is pushing an agenda, if it's trying to cast doubt on a candidate, or trying to "inform" them of something bad or good, then it's a push poll, and I think the WP page made this pretty clear.
Several small scale push polls done in search of or in preparation for a large scale attack are exactly the kind of poll that count as both push and poll. Your example is demonstrating exactly the case where the pollers care about the result, and they're pushing at the same time. The intent is to influence the voter, and also measure how influenced the voter was. That's more of less the definition of a push poll.
