In the MCU, Captain America is referred as the first Avenger but I recently watched Captain Marvel, in which Nick Fury met with Captain Marvel for the first time, even before any other Avengers. In the last minutes of the movie, Nick Fury can be seen to prepare a draft for Avengers Initiative (he had first written Earth's Protective Initiative) which implies that the Avengers started with Captain Marvel. In this case, should Captain Marvel now be referred to as the First Avenger? Am I missing something?
Events in Captain America actually takes place in WW2 era, which clearly pre-dates modern day. even the 90s.
In original team (in MCU), Cap was the oldest member except Thor. While Thor being oldest, he doesn't have any connections to US based activities, but Cap has been working for US government since early 40s and was avenging. You has to have connection with either SHIELD and/or US based organization to be called an Avenger. Group credits Cap's previous heroics and activities and considers him being First Avenger.
If we give a look at character introductions,
Captain America is a fictional character, a superhero who appears in comic books published by Marvel Comics. The character first appeared in Captain America Comics #1 (cover-dated March 1941), from Marvel Comics' 1940s predecessor, Timely Comics.
Thor first appeared in Journey into Mystery #83 (Aug. 1962) and was created by editor-plotter Stan Lee, scripter Larry Lieber, and penciller Jack Kirby. The mythological Thor had appeared previously in Venus #12-13 (Feb-Apr 1951).
The Avengers is a team of superheroes, appearing in comic books published by Marvel Comics. The team made its debut in The Avengers #1 (Sept. 1963),
Carol Susan Jane Danvers is a fictional superhero appearing in American comic books published by Marvel Comics. Danvers first appeared as an officer in the United States Air Force and a colleague of the Kree superhero Mar-Vell in Marvel Super-Heroes #13 (March 1968). Danvers later became the first incarnation of Ms. Marvel in Ms. Marvel #1 (cover-dated January 1977)
We can see that Cap has been avenging since 40s and was a founding member of the Avengers since it's introduction at 63, while Captain Marvel was born at 77. I believe this also makes Cap is the First.
If we take a look at MCU, When Nick Fury activates Avengers Initiative, his original team included Rogers, Stark, Banner, Barton, Romanov, but not Danvers. So if we compare Rogers and Danvers, Steve is the actual First Avenger. How he became First Avenger from all, is that he has been a hero from long ago and considered what he did as avenging.
Captain Marvel merely gave Nick the idea that Earth needs protectors from aliens and other beings with superhuman powers. Captain Marvel wasn't even on Earth until after the events of Infinity War had occured. This means that Nick could not even have asked her to join the Avenger Initiative.
Also Captain America came to be known to the world before the events in Captain Marvel (1942 vs 1995). Therefore one could argue, that Steve is the rightful first Avenger, simply based on the time of the events. This Quora article supports this with some further explanation.