Account, then Security, lists the sign-in methods your account can use. Each one has a Connect button, and once connected a Disconnect one.
Before you disconnect one
Set a password first. The page will tell you the same thing: the point is that you always keep at least one way in, and removing your only sign-in method would lock you out of your own account.
Disconnecting Apple asks for one thing more. If you chose Hide My Email when you first signed in, your login email is an Apple relay address, and Apple may stop forwarding it once Apple sign-in is gone. So the page asks you to change your login email to an address you check before Apple can be disconnected. See Change your password for how.
Connecting it again later
Disconnecting Google or Apple removes it from LoyalPack only, and you can connect the same account again whenever you like. Log in with your email and password (or a one-time email link), open Account, then Security, and press Connect next to Google or Apple. Finish at Google or Apple as usual, and the page confirms it is connected.
Pressing Login with Apple or Login with Google on the log in page will not do it on its own. Once a sign-in method is disconnected, LoyalPack deliberately no longer knows which account it belonged to, and it never guesses from an email address, so the button can only tell you it could not sign you in. Connecting from inside your account is what tells LoyalPack the Google or Apple account is yours again.
If the page says that Google or Apple account is already connected to another LoyalPack login, it is connected to a different account of yours. Log in to that one, disconnect it there, and then connect it here.
On the counter app as well
The Login with Google and Login with Apple buttons on the counter app work the same way as the ones on the website. Tapping one opens your phone browser to finish at Google or Apple, and then hands you back to the app signed in. Connecting and disconnecting is still done on the website, under Account then Security.
Connecting a provider does not change your login email or your password. It adds a way in, it does not replace one.