"Social accounts unavailable"

The Recovery partition is a rescue partition available only under Mac OS 10.7 when you start with the Alt key pressed. I have already explained this to you here:
https://forums.commentcamarche.net/forum/affich-24472174-obtenir-l-image-disque-de-l-installer-lion
You may not have read the answer correctly.

No installation disk? That's not really normal, since these CDs or DVDs, provided with every Mac up to version 10.6, must always be returned with the Mac whenever ownership changes.

However, if you read my previous link, you noted that there is a method of repair without a CD. This method involves the command fsck -fy; again, perhaps you didn't read the page correctly?

Good evening,

The Mac is probably trying to start from a network.
Try starting while holding the Alt key to select your internal disk as the startup disk.
Then, go to System Preferences / Startup Disk and designate your internal disk as the default startup volume.

Have a good evening
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Always specify your EXACT CONFIGURATION: Mac model and Mac OS version!
Francis Sanspseudo - Intel iMac running OS 10.7 + iMac-G5 running Mac OS 10.5

Thank you for this quick response, but unfortunately I have already considered this possibility and it still doesn't work... I am currently starting my Mac from a bootable DVD-RW on which I have burned DiskWarrior, a software that allows me to repair my internal hard drive even though I'm not convinced that my problem comes from there...

Indeed, the problem doesn't come from there, I now have a new MacBook but I would still like to recover the files on my old one... Does anyone have a solution? Please.

After re-reading the initial message, if the problem is a password issue, you may simply have the Caps Lock key on while typing?

If you need to perform a repair after a crash, see this page:
https://www.commentcamarche.net/faq/15279-entretenir-et-reparer-mac-os-x-10-6-et-versions-anterieures

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Always provide your EXACT CONFIGURATION: Mac model and Mac OS version!
Francis Sanspseudo - Intel iMac running OS 10.7 + iMac-G5 running Mac OS 10.5

No, I obviously tried all the possibilities before looking for the cause of the problem or trying to fix it; it is certain that it is not a password issue...

Make a repair, preferably from the installation DVD if Mac OS 10.6, or after starting with the option key pressed if Mac OS 10.7 to access the HD Recovery partition.

It is indeed a Mac OS 10.6 but it’s a Macbook that was given to me by someone who was already using it before, and she didn't give me the installation DVD, which unfortunately seems to be my only way out....

My second solution for retrieving my files would be to connect my two MacBooks with a FireWire cable, but I don't have one and I'm wondering if it's really worth spending €20 to recover the few images, videos, and movies that are there... Especially since I'm not sure it will work.

The target mode, with a Firewire cable, works without issue. You need to have a Firewire port on each Mac for this.
Leave the master Mac on, and connect the problematic Mac when it is turned off. Then start this Mac while holding the T key until it appears as a hard drive on the master Mac's desktop; after that, it is treated like a simple hard drive.

However, you don’t mention whether this sick Mac is running Mac OS 10.6, 10.5, 10.7...? or if you have the installation disk; in short, we know nothing about the configuration.
We don't know if you’ve tried booting from the installation disk, from the Recovery partition, etc... no information at all...

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Always indicate your EXACT CONFIGURATION: Mac model and Mac OS version!
Francis Sanspseudo - Intel iMac under OS 10.7 + iMac-G5 under Mac OS 10.5

The sick Mac is running on 10.6 and as mentioned just above, I don't have the installation DVD. What is the Recovery partition?

A Macbook given by a family member... and already a new Macbook... great all that,

So, just have to wait for the person to return, or for this person to send the DVD, or even buy the DVD:
https://www.apple.com/fr/shop/refurbished

Unless target mode...

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Always specify your EXACT CONFIGURATION: Mac model and Mac OS version!
Francis Sanspseudo - Intel iMac running OS 10.7 + iMac-G5 running Mac OS 10.5

Buying a new MacBook isn't "great," €1450 still hurts quite a bit... It's a necessary evil, in a way.
Anyway, whether you doubt the origin of my sick MacBook won't solve my problem; I think I'll turn to target disk mode.
In any case, thank you for wanting to help me.
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Mac OS X Lion 10.7.3 (MacBook Pro)
Processor 2.8 GHz Intel Core i7

Don't blame me too much for being suspicious, but I notice a surprising number of people who have lost their passwords and don't have installation CDs. Their number is increasing every day right now.
As I am also looking for a second-hand Mac for my wife, I'm checking classifieds, and I hardly ever come across, when the quality/price ratio seems correct, anything but Macs of very dubious origin, or outright suspicious. It's quite incredible.
Or else there are "Breathtaking offers" that are more expensive than brand new equipment.

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Always indicate your EXACT CONFIGURATION: Mac model and Mac OS version!
Francis Sanspseudo - Intel iMac Mac running OS 10.7 + iMac-G5 running Mac OS 10.5

Indeed, I can understand, in my case the password is not lost, I know it by heart, I use it dozens of times a day, as I said the problem is not there....
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Mac OS X Lion 10.7.3 (MacBook Pro)
Processor 2.8 GHz Intel Core i7

In this case, the repair should restore the situation. Unless there is severe corruption. And then, you definitely need the DVD to reinstall.

Have a good evening.
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Always indicate your EXACT CONFIGURATION: Mac model and Mac OS version!
Francis Sanspseudo - Intel iMac Mac running OS 10.7 + iMac-G5 running Mac OS 10.5