When I found out that the Motte and Bailey Fallacy was a known and described thing, I had already noticed it and came up with my own name for it: gecko tailing. Most geckos and other lizards will drop their tail if a predator grabs it. The analogy doesn't quite hold, because geckos don't plan to lose their tails, but it was a pretty cute and clear metaphor in my mind.
The motte and baily fallacy is a rhetorical form of bait-and-switch, where the moment a claim is refuted, disproved, or called into question, the person who advanced that claim will suddenly reveal that the refuted claim was not the main point at all. If people accept their initial claim, they win. And if they refute it, they don't lose. Rather than explaining the concept more, I will use two concrete examples I have encountered over the years on two internet sites (Reddit and Somethingawful) that I have experienced as sites of bad faith arguments.
In the first example, when I mentioned in a reddit thread that there were no permanent puma populations east of the Great Plains, another user quickly commented that there were "hundreds of sightings" from "trail cams" that could be easily seen. When I went looking, I found that there were confirmed reports of vagrant puma in the Eastern US, but that there were not many of them, and most of them were close to their western range, in places like the Upper Peninsula of Michigan. Sightings in places like North Carolina were often dubious at best. When I asked about this, and asked to see these hundreds of confirmed photos, I was told by the poster that he was much too busy to do my research for me. But of course, if there were many obvious pictures that "research" would be a very short google image search. So suddenly the claim had shifted from "This is obvious" to "If you spend hours on research, you might be able to find ambiguous evidence".
In another example, on Somethingawful, someone (probably a tankie) wrote that John Dingell, the former Dean of the House and a popular twitter firebrand, had "advocated the carpet bombing of Iran". When I tried to track down that statement, it turned out to be that Dingell had signed a letter saying that he supported sanctions on Iran for pursuing nuclear weapons. And the letter had been signed sometime before the actual sanctions had come into effect. So the "logic" went something like this: "carpet bombing is detrimental to Iran, sanctions are detrimental to Iran, therefore sanctions are carpet bombing". Now the person who said that was an obvious troll, but it shows the same strategy: make an outrageous claim, and when questioned, retreat to a more modest claim, but then assert that they are essentially the same thing. Bonus points for a sense of indignation that other people just aren't abstract and sophisticated enough to get.
These are just two examples, and of course I don't remember every detail from every internet argument I have ever gotten in, but these were pretty classic examples of Motte and Bailey arguments, and probably the reader can think of many more such arguments they have seen online, as well as in other places.