I use both but the Ngram result is interesting.
My feeling is that 'one way of putting it' is more common. This Ngram seems to support this feeling.
No, it doesn't mean that. It means that there are different ways to essentially say the same thing. This does not contradict 'emsr2d2's description of possible nuanced meanings.
That's the way also works.
Using "That's one way to put it" often implies, in a slightly sarcastic way, that the previous statement was unexpected or that it was not the way most people would say it.
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