Not exactly.  The House forwards the articles of impeachment to the Senate.  The rest is a good summary that you gave.  By "holds the impeachment trial" I meant generally that that is where the trial is held - physically (although its not technically required to be in the "Senate chamber", but where ever the Senate would like to hold it).  The Senate as a body "hosts" the trial is probably a better way of saying it.  The main thing is that the Senate decides whether to impeach, the House decides whether there's cause to impeach.