I would say you are right. Middle C, piano wise is the reference I have used all my pretty long life. I started out on piano, so got that long standing reference during intial theory and orientation lesson. I did have a pretty good idea, as my mother played piano and my dad was a professional reed player. I have never heard 3rd space C referred to as "middle C" by anyone.
Playing the trumpet I have always thought of "low C" as Bflat concert, at the middle C piano pitch. High C was the C above " G" located above the treble clef staff. The one that many players have trouble hitting well, including me. To me, double high G would be the G above high C, and up the scale to the double high C, 3 octaves above middle C, definitely in screamer land. Instruments with different ranges may have a different pitch points of reference octaves below or higher than piano middle C.
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Dave Rafferty
07 Barrington Trumpet
55 Besson 8-10 trumpet
52-55? Roth by Reynolds Cornet
73 Olds Ambassador Cornet
? Czech small bore Trumpet
Getzen 80 cornet
66 Olds Ambassador Trumpet
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