Recognizing `3+x*(2+y)` is compilation - even if the program is being executed while compiling it.

You can continue to argue the point but that goes against every single definition of compilation that exists. Compilation is a transformation of programming language into another form. For example, taking `3+x*(2+y)` and transforming it into a series of byte codes, machine language instructions, ASTs, or even C code would be compilation.

The BASIC interpreter doesn't recognize `3+x*(2+y)` nor does it compile it instead it evaluates that expression using a pair of stacks. You've expanded the definition of compilation to cover almost all computation. It's compilers all the way down to the electrons.

Well, all my compilers use recursive descent for expressions, meaning the stack is used to maintain the current state. Whether you evaluate it while doing this or produce an IR is a trivial difference.

It might be a trivial difference but it's literally the thing that makes something a compiler. It should make sense. How a piece of software works doesn't make it a compiler or not; it's input and output of the software that defines it as a compiler. That's true of almost any broad category of software.

No problem calling it parsing, but yeah, "compilation" feels like a huge stretch. And they didn't do recursive descent - just tokenizing for compactness (when you only have 4k or 16k of RAM you do things like that) - you could still get syntax errors at runtime. In some interpreters it also served to normalize abbreviations to save typing.