The first (turing complete) computer actually build was the Z3.
Not many people know about it, because central Berlin in 1943 was not a healthy place for a computer.

The ENIAC was before that and Charles Babbage's difference engine was before the ENIAC, though I don't think he actually got it out of design

Depends entirely on the definition of computer.

There are so many "inventors of the first computers" it is ridiculous. Almost like creating a complex machine like a computer takes a whole many inventions and people who worked on it over a time span of multiple generations.

Turing wrote the third (and final) program for the Baby.