Mozart's Coronation (No. 26): Allegro vs. "Allegro brillante"

Factual Questions
1

There are apparently two versions of Mozart’s Piano Concerto no. 26, movement I, one with the tempo “allegro” and one with the tempo “allegro brilliante.”

What is the difference? The “brilliante” version does sound somewhat more grandiose.
But the Wikipedia page for Mozart’s K.537 says that the tempo of the first movement is simply “allegro” and gives no indication of where “allegro brilliante” may have come from. Could an editor simply have put “allegro brilliante” there?

You can listen to both versions on Spotify.

2

That’s “brillante,” and it’s a mood marking, separate from the tempo. It’s Italian for “sparkly.” Have you seen this on sheet music, or only recordings? Because if it’s the latter, I suspect that it’s the particular performance that is labeled “brillante.”

3

Here’s the score. The first edition is marked simply as “Allegro”, at least in this edition.