Haisaki

I think the system doesnt matter

In my case my psn email was different to my epic email. I reccomend you to check it into you epic account settings.

Well, thats strange. I had the gray screen that you attached, but after that I login with the first screen that appears (the white one where you can put email and pass or use psn xbox etc.) and with that it worked for me. What happens when you put your email and password? Does the gray screen appears or it just say you cant login?

Are you sure? Yesterday I changed my Epic password because I never used it to log in (I alwas used my PSN account) and it let me in using the associated email and pass.

The PSN login is bugged in Launchbox. I had it, I recommend you to log in to your epic games account in a browser and read the linked email to it, then log in using the email and password, NOT PSN.