For clarity perhaps, I mean Olórin before he was Gandalf.
Looking at the text, it reads Olórin said this or that, while the name Mithrandir is only added parenthetically to note who Olórin is -- in my opinion because from JRRT's perspective, Olórin might be an easily forgotten name, especially before the constructed Silmarillion was published, and these fancy internet days.
In one of the Glorfindel texts, it's said.
"That Olorin, as was possible for one of the Maiar, had already visited Middle-earth and had become acquainted not only with the Sindarin Elves and others deeper in Middle-earth, but also with Men, is likely, but nothing is [> has yet been] said of this."