Does "[card name] or another [type]" require [card name] to be in the proper category?
The rules do not explicitly cover this, but in Magic card templating, the phrase "another [characteristic]" is consistently used to mean "an object that is [characteristic] and is not this object". So, in all of these cases the ability will trigger whether or not the object itself has the mentioned characteristic.

In general, this doesn't come up in rulings, but Ephara, God of the Polis, for example, has an ability that refers to "another creature", and there is no indication that it only functions while Ephara is a creature.