What is the difference between these ring oscillator topologies? How can I simulate one without "noise voltage?"
Your "infinite gain" way of understanding only works for small signals. If that wasn't the case (it would always apply even when the signal is large) then the amplitude of the signal would reach infinity as well. Obviously, that's not what happens.

What does happen is that as the amplitude gets larger the amplifier stages start to saturate and because of that saturation their gain decreases. When a stable oscillation is in place, the loop gain will be equal to 1 (one).

The circuits that you don't quite understand aren't any different from the one you do understand. The only difference is in the implementation of each amplifier. The amplifiers simply use a different circuit. The differential circuit is also very similar only the signal isn't single ended (like in the other circuits) but differential. That means the signal has a copy that's the same but it has an inverted shape (180 degrees phase shift).

You ask about "noise voltage" but I think you're confused about the way how these circuits start in the real world. In the real world there's always noise present and that noise is enough to get the start up the oscillator.

In a simulation this is not the case, the simulator will be happy to find a stable solution where the circuit isn't oscillating. Like for the 3-inverter ring oscillator it will find that all signal nets are at a DC voltage close to Vdd/2 and that that is an OK solution. Theoretically it is but it does not work like that in the real world.

So to get the oscillation going in a simulator we need to "disturb" that wrong stable situation. I usually do that by inserting a pulsed source that can be a current source or a voltage source, it does not matter. I make that source deliver a small pulse like a 100 ns pulse of 1uA (or 10 mV). That will then start the oscillation.

Noise simulations (actually: Phasenoise) are the next step when you have fully understand how to simulate oscillators.