I was looking at the MIDI specifications on this website but I could not find the official specification that maps the MIDI note names to the MIDI note numbers.
Different websites provide different mapping.
For example this website maps C0 to octave -1
while this another website starts mapping with Octave -5.
Is there a specification that we can consider as official ?
There was a similar question here not a long time ago.
The short answer: no, there is not an official MIDI standard, and there is no consensus among manufacturers and neither among musicians.
If you ask for my opinion, I prefer the Scientific pitch notation.
Previous discussion appears half way down page 5. Please check back to that. Includes various links to other relevant info. May answer all your questions.
Geoff