App: Use AUTH_<PROVIDER>_MODE for auth requests

[joselcvarela]

Describe the Bug

In #21239 it was introduced a session mode which uses a cookie to keep user logged in.
Although this have broke some behaviour we had previously.
For example, when project is using Seamless SSO it may be using cookie for AUTH_<PROVIDER>_MODE.
But the user will not be able to use the App as it is strictly set to session mode.

In order to fix this, App should use the mode that is set on AUTH_<PROVIDER>_MODE

To Reproduce

Setup some SSO mechanism and then set AUTH_<PROVIDER>_MODE to cookie.
Now it will not be possible to login into the App, using the SSO provider

Directus Version

main (10.10.4)

Hosting Strategy

Self-Hosted (Custom)

[paescuj]

Duplicate of #21836, will be fixed with #21874 👍