"Right" is what gets the job done.
There is NO right way to install a tube, only a bunch of elements that have to be dealt with (no twists, straight valve, tire seated, etc.) However anyone prefers that gets it done properly is the "right" way (for them).
I actually use either method depending on the bike, removing and stuffing the tire on my 1.9" wide commuter tires, but leaving one side on on my narrower (and harder to remove) road tires.
Sometimes it also depends on conditions, because I'll generally remove a tire if I have to locate the cause of a puncture, but not if it's plainly obvious before I start.
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