So, what does the word “static” mean in this context, that the binary’s behavior does not change?

Same meaning as in a "static analyzer" for code quality: it works without executing the target program. (whereas a dynamic rewriter could run the application under debugger control, observe its behavior, extract additional meaning from that and make changes on-the-fly)

I imagine it means that it takes the binary file, generates a new binary file, and is done, as opposed to rewriting things at runtime or needing profiling information from a prior run of the binary.

This is correct. Specifically, there's a category of tool called "dynamic binary translators" (including Pin, DynamoRIO, Valgrind, and QEMU) that translate binary code one block at a time, just before it's executed.