Yes, but you also make a distinction between a gate with an inverter added at the output and a gate with an inverter added at one input. There seems to be no logical distinction other that perhaps some appeal to symmetry.

As

@BobK points out, there is a simple list comparing all possible functions of a 2 input gate. This ignores the implementation of the gate, and can then be used to exhaustively determine all the possible universal gates. It turns out that all you need is a gate which has one output state that can only be achieved with a single set of input states.

As an example, consider a transmission gate with a down on the output. It implements one of the 16 functions and one that is universal. The function is NAND.

Change that pull down to a pull up and look at the logic function. It's

__not__ NOR.

The only reason why NAND and NOR (and AND and OR) exist while these odd gates don't is that they may be easier to imagine, and can be more simply expressed in

Boolean algebra. However, that's a failure of our thinking and of the available Boolean operators. The XOR function is not one of those "simple" gates (and neither is it universal) but it has it's own symbol in Boolean algebra. Dear Morgan's law is different for XOR, why can't there be a gate with different Associative, Commutative, and/or Distributive laws? When handling XOR we often break it down into an equivalent using AND, OR, and inversion, the same could apply to some other new gate type symbol for the new gate type "left inverted OR" or perhaps LOR?

Once we have this, then we find that LOR is simpler than OR in silicon, and can be manipulated just like AND and OR in Boolean algebra, and can be used in its own to implement any other logic function.

How does it now differ from NAND or NOR (other than being ugly)?