I don't think it is ash - the bark does not look right. Having a cross section showing the face grain and end grain is often helpful to identify wood from a picture.
Yes, those terms match your pictures.
My vote is for beech, but it's still a guess. It grows in the region (Jura Mountains), is a hardwood and has a similar grain pattern and bark.
From the trees that grow there, beech is completely a possibility, but I can't find the little dashes that I usually see on the edge grain of beech wood, thus I eliminated that possibility. Still I don't know if every beech look the same, so, maybe you're right?
Oak, ash and maple are all other possibilities for that forest, but maple was very unlikely, thus the other guesses (oak and ash) are completely plausible too.
Do you think of a test, or anything else I could do on the wood to help further determine the essence?