I think it will work much better if you backup the Thunderbird folder, which contains the Profiles subfolder and the all-important profiles.ini.

C:\Users\username\AppData\Roaming\Thunderbird

Copy the Thunderbird folder into the same location on a new or existing computer, depending on whether you're transferring or restoring the TB setup.

https://support.mozilla.org/en-US/kb/moving-thunderbird-data-to-a-new-computer

I would also recommend giving the profiles distinct names, instead of 'default', using Profile Manager.

In the Thunderbird folder that are backed up by my online backup service there are only these tree profile folders in it, and there are no profiles.ini contained in it. Is that the reason for my problem maybe? I find it difficult to understand the role of "profile manager", I understand that it can make profiles and that one can choose which profile should be used when starting up TB. But exactly what to do with it I dont understand. If it can just rename the profile folders it would be much easier to do it directly. Since I already have the profile folder there do not seem to be any idea in creating any. And since it has to do with what profile are used when TB starts I experimented with writing the names of the 3 profiles after pressing the button "create profile" which is probably a totally stupid idea, but I think it was something like this I did some weeks ago when I had exactly the same problem and suddenly it worked. But what happens is just that the mail that are stored at the server is loaded into TB, and not all the 3-4 GB of mails that the folders contain, and not the complete folder structure, but just the standard structur for one account.

Renaming the profile folders in Explorer wouldn't work unless the names were also changed in profiles.ini, which would be done if you renamed through Profile Manager. But if you have a profile folder that isn't recognized by TB, I think it would work to create a new profile with PM, and browse to the location of that profile folder.

http://kb.mozillazine.org/Moving_your_profile_folder_-_Thunderbird#Use_the_Profile_Manager_to_move_your_profile

Thanks, I will try to do that. But what do you mean by "browse to the location of that profile folder"? In what should I browse, in the PM? I simply do not understand how all of this works, if one choose a profile to open TB in PM does that mean that the other profiles in the profiles folder are passive ? Or are all these tree profile folders that I have in my backup necessary for getting all of my accounts and preference to show up? As you can see I have no idea about how TB function. Thank you for your help.

The terminology is probably a bit misleading. When you 'create' a profile in PM from an existing profile folder, you are really 'registering' the existing profile so that TB recognizes it as a readable profile. TB only reads a single profile at once, in most cases.

When you 'create a new profile', the next screen has a Choose Folder button (see picture). Click that button to browse to a profile folder that already exists, but is not currently listed when PM starts.

Thanks Sfhoves !!!  :-))))

This solved my problem. From I read your last information it took me a minute or two to have all my accounts and setting exactly as when before my old computer crashed. And ALL my precious mails. I was at the brink to give up after struggling with this for a week. Thank you for responding so quick to my questions. I am grateful, thank you!