Your question is somewhat unclear because there is no clear definition of what is and isn't a transpiler.
However, I suspect the distinction you are thinking of is that a compiler produces a program that runs natively on your computer whereas a transpiler produces a program that does not run natively but instead depends on another language.
For example, Language X is transpiled into Javascript, and Language X
traspiler/compiler is written in Language X but it always ends up
being javascript before it runs... so is that still bootstrapping?
I do see what you mean, Language X isn't totally independent of the other language, it still requires JavaScript to run. If the Language X compiler produced native machine code it would be fully independent.
But the term bootstrapping in programming languages refers specifically to the language compiling itself without concern for the target language. So, yes, as the term is generally used, such a language is bootstrapped.
It's so confusing specially thinking about what happens if Language X's syntax changes after it's bootstrapped, would the traspiler need to be re-written?
Firstly, the compiler vs transpiler distinction does not matter here. In either case, introducing syntax changes into a self-compiling language is a bit tricky.
Yes, the compiler has to be changed to support the new syntax. Of course, it would be very painful if the compiler constantly had to be rewritten with every change. Generally, languages are not self-supporting while they are initially being developed. Only once the syntax is relatively stable is a self-compiling compiler implemented. Once a language is more stabilized, most features are backwards compatible, allowing old code to continue working. Thus, it is isn't necessary to rewrite the compiler constantly, it can just use the same syntax.