Two substantially different but similar cases: Are they the same in terms of table cadinality

Some constraints are best enforced in the database, and some are best enforced in the application.

Constraints that are best enforced in the database are usually there because they are fundamental to the structure of the data model, such as a foreign key contraint to ensure that a product has a valid category_id.

Constraints that are enforced in an application might not be fundamental to the data model, such as all FooBar products must be blue - but later someone might decide that FooBars can also be yellow. This is application logic that doesn't really need to be in the database, though you could create a separate colours table and the database can require that the product reference a valid entry from that table. BUT the decision that the only record in colours has the value blue would still come from somewhere outside the database.

Consider what would happen if you had no constraints in the database, and required them to all be enforced in the application. What would happen if you had more than one application that needed to work with the data? What would your data look like if the different applications decide to enforce contraints differently?

Your example shows a situation where it might have been more beneficial to have the constraint in the application rather than in the database, but perhaps there was a fundamental problem with the initial data model being too restrictive and inflexible?

Yes, I believe you are on the right track. The gameDate and gameScore fields are more a part of the game entity and not the player, so they should be on the Gametable. Besides, bridge tables should really only contain the related ID fields.

If gameDate and gameScore were on the PlayerPlaysGame table, they would only be dependent on the gameID and not the playerID. By definition, this would constitute a partial dependency, and would violate second normal form.