Thanks for your help, horiatb!
Actually I just succeed doing to "upgrade", reinstalling PS 1.7.1.2 from scratch, copy back the images and small modif. I made on the PS 1.7.1.0, using the PS 1.7.1.0 database with the latest patched PS 1.7.1.2 version. Everything seems ok! (I just needed re-doing some setup)
You just need to update these 4 entries from the previous databas into .../public_html/app/config/parameters.php
DB, USER, PWD, TABLE_PREFIX.
Of course, it worked just because it was between 2 same minor PS 1.7.1.x versions (it will not work with 1.7.2.x).
Anyway, I found the Prestashop releases/update process A SHAME, and will not upgrade anymore for the 5 coming years (already skipping the next major release), and never upgrade anymore with a major version newer than 10 month!
There is ZERO support and compromise with Prestahop, and we are just losing time and energy...