You seem to be all over the place here. So Trump sending Giuliani to talk to Ukraine is a different matter from Trump using Barr to do the same thing? Even though Giuliani claims he was directed by DOJ as well? This is getting nonsensical.
It depends on the subjects they're talking about. AG Barr and District Attorney John Durham have been engaged in an investigation into the U.S. intelligence community and the 2016 election. Ukraine is involved, as apparently they have important evidence, and that was the first topic on corruption that Trump/Zellenskyy touched on in their call. Giuliani should not be involved in this at all, and there's no reason why Barr would allow Giuliani to be involved in this, since he firewalls DOJ investigations.
Giuliani is representing Trump on the related matter of 2016 election interference, and some actions that Democrats in Congress took towards Ukraine are relevant to ongoing congressional investigations. Hence, some evidence that Ukraine has is relevant to Giuliani as well.
What matters in the impeachment inquiry is the degree to which Trump is mixing state business and personal campaign research. If the Dems can get someone like Giuliani to brag, "We ain't sending aid to Ukraine unless they give us the dish on Biden," then Giuliani muddies the waters and is pretty much the smoking gun the Dems need. This is the conclusion that the Dems like Schiff want to get to, but so far the evidence is only circumstantial.
The DOJ is likely going to issue a report that says the U.S. intelligence community, the Democrats, and some members of the media purposely carried out a counter-intelligence operation against Trump to undermine him, beginning in the spring of 2016. That report is then going to play heavily in Trump's 2020 campaign. Trump is going to say the Dems are corrupt, including Biden (if he's still in the picture).
However, Trump shouldn't pressure Ukraine to investigate the Bidens. He should sit back and let the DOJ do its job, or just go with the information on Joe and Hunter Biden that's already been out there for years.
It does look like, from the call transcript, that Trump is inviting Zellensky to voluntarily dish on the Bidens, but he is being careful to be a bit vague and indirect. Zellensky is saying that he will expose corruption in Ukraine and bring in a state prosecutor he has 100% confidence in. However, the phone conversation is inconclusive. Trump raises the subject, but doesn't make a concrete ask. There's no threat or demand for direct quid pro quo. It goes without saying that the U.S. has the ability to send a lot of money to Ukraine, and it's clear that Trump has a lot of influence with the Republican Senate. It's clear that Zellenskyy wants to make the U.S. and Trump specifically happy. But you can't impeach a president for conducting foreign relations. That's not corruption in and of itself.
The standard of High Crimes and Misdemeanors is a pretty high standard to reach for impeachment.
