Just going to take a step back here.
Let’s do this methodically.
Yes I agree it sounds like a LF card.
However, can we give you a few things to do / try.
I understand you are trying to do it on a budget, so I will take that into account.
I know you have tried RC522 rfid reader.
Have you read any other cards with your RC522
Generally, if a card has a rectangular antenna it is likely HF, if it is circular it is likely LF. Yours looks rectangular.
?
Are you iPhone or Android? Either way, Could you:
Download TagInfo by NXP
Turn on NFC on your phone
Remove any phone cover
Open TagInfo
Place your card on the back of the phone, and let us know if it reads.