I can't understand/see the difference between a push-pull and a totem-pole output.
Do these circuits have a transistor in the input for something like "double amplification?"
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You'll notice in the push pull stage that you need a PNP and an NPN transistor, whereas the totem pole driver uses only NPN transistors. This is useful because NPN type transistors are usually easier to make, and support higher current for a given size than PNP type transistors.
To address your question on "Double amplification", a push pull driver doesn't necessarily give more amplification, but is used because it is more efficient than a single transistor amplifier because (ideally) only one transistor is on at a time, and all the current goes through the load.