Memory Cards with old Sony consoles are a bit of a pain in the ass, because besides just plain badly made third party cards, Sony also forced low storage capacities, to sell more of course.
some developers took it too literally so even the official 16MBs don't have 100% compatibility
As far as I know from reading years ago, that wasn't something that happened "on accident". Sony intentionally forced certain companies, usually big ones such as Square/Enix, Capcom and Konami to make checks on the memory cards to see if they were from Sony. If not, they'd refuse to save or create fake, corrupted save files. This was most easy to find with 16MB and above cards.
God Hand from Capcom and one of the Silent Scope games from Konami are two games I know first hand, but I know there are more, and also happened with PS1 games. It's... Very
nice from Sony, but what do you expect from a company that doesn't even let you format your memory cards without a third party involved or freaking homebrew...
As of today I have one original PS1 and PS2 MC, and also two third party ones for each too. I've yet to find any issue with the third party PS1 one, it actually works pretty damn well, looks exactly like the one in alexander1970's photo and I bought it from aliexpress, which is not your "100% quality" online store at all , but I'm sure there's quite a bit of a lottery with these aftermarket things.
What I'd recommend you is to have a way of launching
FreeMCBoot and make backups of your save files from time to time. You can transfer PS1 saves from a PS1 MC to a PS2 MC through the systems native menu, and from there to a USB stick through
ULauncher. That's what I've been doing since I got a PS2 again, both to preserve my saves, and because of the horrid sizes of the memory cards not letting you save many games at once, specially on PS1, 15 slots are not enough one quite a few games can take more than one.
I don't recommend getting a third party PS2 MC that's not 8MB just to be safe. That extra space sounds great but the way most of them achieved it wasn't safe at all, using things such as compression that would lead to corruption of saves.
I imagine you also know that with "128Mb" they mean Mega
bits, so in reality it's 16 Megabytes, which mind you, if they worked, 16MB are quite enough (unless you play sport games, those can take huge amounts of space, some even half of a 8MB card).
If you try to buy official cards, pray that they've not seen too much use, because wasting time to find one only to be worn out and failing sucks tremendously.
All this is so much less complicated on the GameCube... I've got two third party generic 128Mb cards for who knows how many years and they still work perfectly, and on GC the size of the card doesn't cause any issue whatsoever, I have dozens upon dozens of saves on each and still lots of free space.