Four words worth considering together are spin, pivot, talking point, and weasel word. From M-W:
spin (v): to present (information) with a particular spin, e.g., spin the statistics
spin (n): a special point of view, emphasis, or interpretation presented for the purpose of influencing opinion, e.g., put the most favorable spin on the findings
pivot: to turn on or as if on a pivot
talking point: something that lends support to an argument
weasel word: a word used in order to evade or retreat from a direct or forthright statement or position
Politicians, religious leaders, ideologues, partisans, and others (not to imply that these categories are mutually exclusive or exhaustive) often present their arguments in terms of talking points incorporating a particular spin and replete with weasel words. When called or pressed on that spin, they pivot back to their talking points, making full use of the weasel words. What are we talking about mostly in the context of the OP's question? Weasel words.