As I had a lot of trouble understanding the problem, I told myself that keeping track of it on this forum could help other French people in the future. I wasted a lot of time believing that I didn’t know how to configure the MMS module, but the problem was elsewhere.
Information little or not communicated by mobile telephone operators in France is that the size of MMS is limited to 600 KB. Each time I tested a new configuration, by sending a photo, an image or a gif (never the same to help me navigate the different tests), the file was most often more than 600 KB, but not always, which made everything very random and therefore even less understandable.
On the other hand, from the moment I found traces of the existence of this limit of 600 KB on a site which did not provide any source, I was able, with a few tests, to note that this number of 600 KB actually seemed to be the real limit, and then I was able to quickly resolve my problem.
As for solving the problem itself, it consists of manually modifying the file ~/.mms/modemmanager/mms and replacing the line
TotalMaxAttachmentSize=600000 (if you are a gamer you can try 614400 but I preferred give me a margin of safety for the moment.)
I don’t know what this limit is in other countries, but I found French people on different forums who were as lost as me although they had APN, MMSC and MMS proxy information, and who didn’t found no help there, and now I tell myself that perhaps they were stuck on this MMS size limit problem without knowing it.