Currently the only way to check if a name is blocked by Minecraft’s Name Filter is by going on the Minecraft site itself or sending a API request with authentication. In WEB-1382 it is stated that the reason why you need to authenticate to see if a name is blocked is because some names are only available to certain users but names that are blocked by the filter are blocked to all users so I feel like it should be displayed that it is blocked by the name filter without needing authentication. Though this shouldn’t be a high priority since it will mostly only help Account Lookup sites many users still use them so it should still be looked into.
I agree messaging should be clearer. We will look into improving this.