You’re saying that Trump, no longer president, is covered because his alleged offense occurred while he was president. Your reading is not untenable. Yet, the sentence is in the present tense.
In an impeachment of, say, a federal judge, the presiding officer is the President of the Senate (i.e., the Vice President). In practice, this doesn’t happen, and the President pro tempore presides. In the case of a sitting president, there would be a conflict of interest, hence, the Chief Justice. Madison’s notes on the Constitutional Convention are explicit on this substitution. Having said this, the notes aren’t binding. As an aside, it would be inappropriate for the Vice President to preside over his own impeachment, but the Constitution is silent on this.
The Chief Justice, joined if necessary by his colleagues, may have to decide the matter.
I would imagine it's because the Founding Fathers never thought that we would ever try a former president for impeachment. Never imagined someone like Pelosi and Schumer obviously. Trump was president when he was impeached so the dignity of his former office would make it a job for the Chief Justice to preside. But I expect his lawyers will challenge the whole idea of a trial now that he's out of office so the whole question may be moot. We'll see.