If you use Firefox sync and use one login your going to get one overwritten the other data. If you want separate sync account you need separate Firefox sync account with different email and password to keep Firefox profile accounts separate and not merge or overwrite the data from one and another.

I only opened the second account today from a second PC to backup my actual data, as the ones from the first PC got overwritten.

Let’s say I have only a single PC and a single Mozilla account, how do I overwrite (update) the data from my PC to my Firefox account and not the opposite ? For example, I have an old Firefox account that I did not use for a while. I want to update it from my PC with the new bookmarks and addons in order to restore everything if I reinstall my OS or buy a new one. How can I do that ?

Then as stated you need two separate Firefox sync account with different email and password login and then on the PC have two separate "user" account so no two Firefox are the same and separate from the other "Local user" accounts. A old pc not accessed will not be able to sync if the sync log is the same it will replace sync data with the new computer login and the old will be gone. What you do is really backup your firefox profile to insure the data is saved but if that old one doesn't sync then chances that data won't move over. It sync from which device logs in first and is primary computer so if you don't keep pc user account separate and separate Firefox sync accounts your going to loose data regardless as one will replace or over-write the other data when using sync.

Let's put it another way. Usually, with any backup system, you have the possibility to backup and restore, or eventually to export and import data. But with Firefox, I only see synchronization.

Therefore, when I synchronize, I cannot chose to backup up my PC profile to the Mozilla online one or to restore my save mozilla profile online to my PC locally.

Is there a way to choose if the synchronization backs up or restore my profile ?