The $ObjectType global gives you access to refer to elements of your Salesforce org's schema. It does not provide access to cross-relationship data, or any data at all in actual records; it only yields information about the objects and fields in the org.
If your object has a field Relationship__c, you'd refer to its fields in a validation rule via the ...
You can create a custom field in Setup > Object Manager > Activities > Fields & Relationships (in Classic, Setup > Customize > Activities > Fields). From there, use the formula type, a return type appropriate for your field, and the formula would use the field relationship path (e.g. Account.Name or Account.Some_Custom_Field__c).
((field_name = 0) || (field_name = 1)), 'Test 1', IF( ((field_name < 0) || (field_name = 2) || (field_name = 3)), 'Test 2', '') )
Instead of using ,'' for your false return on the first if statement, you want to use your next if statement. That way it reads more like
if first option is false, if second option is false, then ''
Please have a look to following salesforce help link
you can build formula with several IFs like following
IF($User.City = "Napa", 0.0750,
IF($User.City = "Paso Robles", 0.0725,
IF($User.City = "Sutter Creek", 0.0725,
IF($User.City = "Los Olivos", 0.0750,
IF($User.City = "Livermore", 0.0875, null