OP explains all that admirably in the section entitled: "Terminology: BirdNET vs BirdNET-Pi"

I use both. I have a BNPi at home in our summerhouse with a mic on the outside. We live next to a park and the bird song can be deafening in summer!

I have the Birdnet app on my mobile and its ideal when out walking to do a quick survey or identify a song I don't recognise.

Whilst I'm commenting here, I'll drop a shout out for the "Flora Incognita" app for plant identification.