Welcome, Jaren! Hope to see many more thoughtful posts from you.

Yes, it was still called the Corleone Family. Michael turned it over to Clemenza as his caporegime, but Michael ran it. When Clemenza died, Pentangeli took it over--and was still Michael's caporegime. Remember the line in the boathouse: Frankie wants to run "my family without you on my back," and Michael says to him, in Italian, "Your family is still called Corleone and you'll run it like a Corleone."