Putnam's main point was that we can be mistaken about what a thing is. Moreover, that when we can have two things (call them A and B) that we think are of the same type that we can not only be mistaken that A and B are of the same type, but that A could fit the type and B might not.
I think he's making a slightly different point. His point is that the reference of a term, which determines whether, say, the setence "Water is H2O" is true or not, depends on the environment in which that term came to be used. And this could be true even for speakers who were otherwise molecule-for-molecule identical. So just looking inside your head doesn't tell me enough to figure out whether your utterances of "Water is H2O" are true or not: I need to find out what kind of stuff was watery around you when you learnt that term! Which is the kind of surprising bit.
Yeah, this is basically right. Putnam was defending externalism about mental content, the idea that the content of our mental representations isn't fully determined by intrinsic facts about our brains. The twin earth thought experiment was meant to be an illustration of how two people could be in identical brain states yet be representing different things. In order to fully determine the content of my mental states, you need to take account of my environment and the way in which I'm related to it.
Another crazy thought experiment meant to illustrate semanti...
Thagard (2012) contains a nicely compact passage on thought experiments: