Yes, any legally binding consequence of being impeached is contingent on actually being convicted in the Senate trial. (Which is why he got off in the first impeachment.)
The common view among legal scholars, as I understand it, is that being barred from future office (on top of being removed from the current one) is an optional punishment that the Senate can choose to impose, or not, as part of the verdict, if they convict. Presumably they would do so if Trump was convicted after the end of his term. (Edit: The Constitution does not specify how this should be determined, but apparently the accepted practice upon conviction is to immediately take a separate "sentencing" vote on whether to bar the convicted person from future office. It's unclear whether this vote also requires a two-thirds majority, like for conviction, or merely a simple majority. In past cases, involving federal judges, the Senate has applied a simple majority threshold. But a vote that didn't reach a two-thirds majority could potentially be challenged.)
An impeachment trial after the term of office has expired is in itself a novel situation that is not anticipated in the Constitution (though there may be precedents with cases involving other offices, I don't know). You would think that'd mean that Trump could challenge its legality before the Supreme Court, but that's pure speculation on my part.
I must say I'm surprised that it looks like Democrats in the House are moving forward with impeachment, which is only a little bit more than a symbolic exercise at this point, and still seems unlikely to result in conviction the second time around, given the utter shamelessness and lack of patriotic duty demonstrated by most of the GOP (many of the members of the jury are, after all, co-conspirators). The politics of it are uncertain at best, too. Could it really be… a matter of principle? (And anger, I suppose.) Or perhaps it's the argument that unless a clear message is successfully sent that actions such as this will not be tolerated, democracy is absolutely screwed, so there really is no choice but to try it.
