Impeachment as contemplated by the Constitution does not consist merely of the vote by the House, but of the process of sending the articles to the Senate for trial. Both parts are necessary to make an impeachment under the Constitution: The House must actually send the articles and send managers to the Senate to prosecute the impeachment. And the Senate must actually hold a trial.
If the House does not communicate its impeachment to the Senate, it hasn’t actually impeached the president. If the articles are not transmitted, Trump could legitimately say that he wasn’t truly impeached at all.
That’s because “impeachment” under the Constitution means the House sending its approved articles of to the Senate, with House managers standing up in the Senate and saying the president is impeached.
This appears entirely false; the Constitution says nothing whatsoever about the House managers, nor about sending the articles to the Senate. It simply says that the House has "the sole Power of Impeachment", which implies that the House's power of impeachment is something it has in its own right, and not with respect to the Senate. Thus whom the House votes to impeach is impeached. This is the way it has always been interpreted; when the House managers stand up in the Senate and say that the President is impeached, they are reporting what the House has already done, not engaging in a performative act of impeachment.
Feldman goes on to recognize that technically the Constitution says nothing about sending anything to the Senate, but, he says, "the framers’ definition of impeachment assumed that impeachment was a process, not just a House vote." This is perhaps in some sense true, if we are taking 'impeachment' in the broadest sense; Hamilton in Federalist 65 says that the impeachment trial is a "method of national inquest into the conduct of public men". But impeachment itself is just charging someone with "high crimes and misdemeanors", and is distinct from the trial. And while it is true that the point of impeachment is to have an impeachment trial, it does not follow from this that impeachment itself requires anything more than an inquiry and vote in the House.
Contrary to what Feldman argues, the House did not vote "to impeach (future tense)" President Trump; what it voted on was the resolution, "That Donald John Trump, President of the United States, is impeached for high crimes and misdemeanors". And the House voted that yes, he is impeached. There is a future-tense part of the resolution, the resolution that the articles "be exhibited" to the Senate. But that is presented as an additional thing to the claim that Trump "is impeached".
But the fact that Feldman could even think this was a plausible argument is a sign of just how out in the deep waters we are; it's just not heard of for the House to vote for an impeachment resolution and then not send the articles to the Senate for the impeachment to be tried. (It also suggests very strongly, I think, that any attempt by Democrats in the House to go this route will backfire on them.)
