It never did that on my systems.

It would have been useful to know that beforehand, but I didn't even suspect having such a problem. The guy at the store was really helpful and informative with a lot of tips, but didn't mention anything like it.

I don't think I'll open the PC and reformat Win again, just to get rid of that separate partition, but it'll probably come bite me in the ass if that old HDD ever fail/dies. We'll see.
Possibly because the old disk still is Disk 0 for the computer and the SSD is Disk 1. The guys at the store installed the new stuff but they returned the computer still running the old installation. They did instruct me on what I had to do to format and install win 10, but didn't mention anything about win creating a boot partition separately.

I killed it using the Windows 10 installation system, so it was there. I don't think the installation system allowed me to kill those secret partitions, though (I actually don't remember for sure right now).

What I did remember right now is that this PC originally came without a Windows installation, so it was me who installed the win copy that created those two secret partitions.

If you are able to boot Linux (a Live CD works fine for this) and have a spare storage device, you can back up the system and/or OEM partition and restore it later. (There are Windows tools that will work, I believe, but they require extra downloads. Running Linux in a VM or using WSL unfortunately won't work for this, as you can't directly access the host hardware.)

The procedure, once you've booted into Linux and have a command prompt, is as follows:
1. Use the "lsblk" command. This will display a list of the block devices the system recognizes, along with their sizes and mountpoints. This way, you can identify which partition you want to back up. Also, make sure the partition doesn't have a mountpount; if it does, you'll need to unmount it.
2. The command you need to use to back up the partition is:
dd if=/dev/sdX of=/path/to/file bs=4K
where sdX is the device you want to back up (likely sda3, but check with "lsblk" first), and /path/to/file is the place you want to save the backup image. (Chances are the path will be something like /media/sdY1/file.img if you're using a USB device for storage.)
WARNING: Don't mix up "if" and "of" when using this command; doing so could overwrite the partition you're trying to back up.
3. Actually, I lied; you likely need to put "sudo" at the beginning of the command mentioned in step 2. Because accessing hard drive partitions directly is a safety and security risk, only the superuser can do this.
4. If you later need to restore the partition from backup, the procedure is the same, but switch the "if" and "of" arguments of the "dd" command.

It's best to boot off a Linux live disk and use things like gparted to wipe partitions and drives... Windows (especially 10) has a mind of its own and may not be able to delete all partitions or even show them.

As for the rest of it, I concur with what other people have mentioned before: it's probably best to disconnect your HDD during the installation of windows on your SSD, then reconnect it once that is done and partition/format it from your newly installed OS.

This story sounds all too familiar.

For now I'm going to leave it as it is. Though I know that when/if the old HDD fails I'll have to reinstall win again, this machine isn't exactly my main one anymore and any stuff that I had only here, has been backed up before the format.

So I'll just have to be extra careful with what I leave here.

I wouldn't really trifle with Windows after that, if it isn't your main machine. Linux can breathe life anew into an old machine. It isn't all that scary to use, either. Many desktop environments offer an experience similar to, if not outright superior to Windows.

I've used Linux in my last job and it has saved my ass at home once, booting it from a pendrive. I like it, but it's just not an OS that suits my everyday needs. :)