First and foremost, hello, I'm a new member I have posted this question on stackoverflow, however, it was indicated that it was not related to software development enough.
I do not know where exactly to post it. I hope I will find some help here. So, my issue:

So my initial idea was to keep all (!) my files in three external hard drives: One for essentials
(1TB), and the other two - the exact copies of all the files (3TB each). By all files I mean
including my first drawings with CorelDraw when I was maybe 5. So I have bought ADATA
HM900 3TB (formatted as FAT32) portable hard drive (the one that needs the external
power source), and copied all my data from different computers there (about 800GB), both
Windows and linux (I've been using Windows for over a decade and now I'm on linux). I
have done the same with another hdd - Maxtor M3 Portable 1TB (1TB will suffice for now),
formatted as NTFS. I did not care about filesystem differences (including ext4 on my linux
machine), and it all worked well for some half a year.
However, although having used those hdds properly (no physical shaking, no holding
upside down, safe removal, etc.):
1 I started getting input/output errors on my 3TB drive when copying files, mostly, but not
only(!) files containting .git foder. After that, copying process was terminated, and I could not
reach the folder that contained the "bad files". Even after disk restore (the default Windows
one, right click on disk properties) I did not manage to recover all the files. Btw, that was
certainly not the reason of files over 4GB, since I have none of those.
[2] I started getting input/output errors on my 1TB Maxtor hdd (NTFS) with the same
problem, with different folders and files though (there were no .git folders in this case)
[3] On copying files from linux PC to Maxtor hdd today I got a crash (the pc just froze, I
restarted it), after which neither linux nor Windows PCs detect this hdd after plugging it in
[4] I have bought another hdd: ADATA HC500 1TB, and formatted it with ext4. However,
windows PCs could not recognize this hdd (it appared on Control Panel->Disk management
though).
So, after having lost some files and some temper, I am wondering how could this situation
could be improved. Basically, the problem statement is this:
How could I use at least two HDDs (and which filesystems should I format them to) for
identical copies of all my files with the ability to plug them in both linux and Windows PCs (I
still use Windows sometimes), and minimizing the risk of such occurences happening again.

I have a Google Drive folder with all the necessary screenshots:
https://drive.google.com/open?id=1qd2oqegYIGpulU1hces-tAyeGW6TDutX

You can just make .iso images of both systems and then open them when you need some files.

Format it as NTFS and store them there.

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Thank you for the answers. @Spildit, thank you for such a thorough answer! It is a lot to digest. I did not know that older drives, although having smaller capacity, are actually more reliable. I am inclined to follow your advice.

There are also a few things I would like to clarify:

[1] I use mostly Linux with ext4, and the distro does read/write to NTFS, so according to your answer, I should then proceed using ext4 on main drive running Linux, but use NTFS for all the "data-backup" Hard Drives, yes?

[2] Speaking about SATA/IDE, and since I am not in actual need for drive to be both Portable and have USB, it is actually a better idea to use non-portable HDDs, if I understand correctly? (it would be cheaper as well)

[3] The talk about PSU as I understand is the one that is placed inside a PC, yes? I might not understand it correctly. I am backing up from the laptop mostly, so I'm limited to my laptop's battery.

[4] Are all your suggestions valid for NAS? (including quality PSU, hard disk models and so on). Or is NAS just not worth it for my goal?

[5] Pardon me about this one for little knowledge in this area, but as I understand SSDs could be in no way compared to HDDs in terms of reliability/redundancy (HDD being superior), correct?

Thank you for explaining and clarifying!!