If so, then you should be agreeing with me

From wikipedia:
"A transposing instrument is a musical instrument for which music notation is not written at concert pitch (concert pitch is the pitch on a non-transposing instrument such as the piano). For example, playing a written middle C on a transposing instrument produces a pitch other than middle C"
and, under Transposition at the octave:
"Most authorities include this type of notation in the definition of "transposing instruments",[3] although it is a special case in the sense that these instruments remain in the same key as non-transposing instruments."
https://en.wikipedia.org/wiki/Transposing_instrument
If you don't like wikipedia (despite the excellent sourcing), then how about:
https://www.vsl.info/en/academy/woodwinds/piccolo
You are correct that most people don't really think about octave transposition as the same as "regular" transposition, which is why I took pains to say that it's "technically" transposing.