Briefly stated, you have "subpoena" correct and those you argue with do not. I don't know WHY this argument is going on. This is very simple stuff.
As mentioned, a congressional body / committee might issue a request - a letter - and then if they didn't get what they want, they could issue a subpoena.
Even if they did issue an actual subpoena, if it is done outside of an official impeachment hearing, it still is issued from a co-equal branch, and the executive still has every right to exercise executive privilege and not comply. Once the hearing is voted on and official, this executive privilege is severely limited and the subpoena would then have the force of law, allowing it to be ruled on by the judiciary in the form of a suit, which would then give them enforcement ability.
That's correct, and I don't see why people are arguing with you. No right-winger has even claimed the House could not have impeachment hearings against Trump, or whoever. That's part of their job.
But the specter of a small cabal in the House hijacking that Constitutional right has many glaring problems, and few if any advantages, either long or short term.