According to the very quote you posted above, the intercepted communications were between "Russian officials" and "associates of...Trump."  One would expect a lot of "Russian officials" to be tapped, and if they were on the phone to "associates" of Trump, then what those associates said would be heard.  The NYT article in absolutely no way "confirms", as you put it, that Trump was tapped by the Obama administration, let alone that it bugged the phones in Trump Tower.
A basic principle of essay writing I teach my students is to ensure that your quotes actually support the argument you're making.  Yours doesn't.  There is no basis in the quote for saying, "Trump is vindicated that the intelligence community under the Obama administration was wiretapping Trump...and even the NYT article above confirms this."