Another example is Rain Robinson talking about the show "Mission: Impossible". If someone were to watch some of the later episodes of the show, would they notice one of the main characters resembles a certain Vulcan Ambassador?
One indication that Ms. Robinson doesn't live in our "real" universe would be the spacecraft model on display in her work area. It isn't a space shuttle, it's a DY spacecraft.
Yet, if someone in the 24th Century on Voyager were to access historical records on America's 44th president, how much of this would be present?
Other than Washington, Lincoln and FDR, we don't know who the American Presidents were in the Star Trek universe. After the events of the Eugenics War, would Obama for example have been elected to the Senate or the White House?
Were other Presidents mentioned on the show?
In a "hardware" variation of this idea; Kirk's Enterprise was supposedly named after Archer's NX-01. The NX-01 was named for the Space Shuttle Enterprise, it was even followed by an NX-02, Colombia. And, as we all know, the Space Shuttle Enterprise was named for the fictional ship "Enterprise" from Stare Trek.
No, the NX-01 was named after the Enterprise E that Cochrane saw through his telescope, Deanna told him the name of the ship. Kirk's ship was named after a long line of American naval ships.
If the Enterprise (prime) were named specifically
after the NX-01, then wouldn't the NX-01 have been present in the display of former Enterprises in TMP?
Debatably, prior to the events of FC, the NX-01 historically wasn't named Enterprise.