So, Zuse's development were indeed important, but the Zuse Z3 had in important limitation that the article doesn't mention, which leads some people to not count it as "first": The Z3 was largely not Turing complete.
This is because the Z3 had no conditional branching. In 1998 it was shown that the Z3 actually didn't need conditional branching, and it was Turing complete but only because it is possible to unroll conditional statements into a version where both branches are evaluated and then the result of one is cancelled. However, at the time of the Z3's development and use, this wasn't known.
So while the Z3 is indeed an important early computer, it shares a significant discontinuity with modern computing that reflects the time. The Z3 was not intended as a "computer" in the modern sense, it was more of a calculating machine that could perform repeat arithmetic more quickly. It was never intended for the more complex roles we use computers in today.
A lot of people don't count the Z3 as the first programmable digital computer because of this limitation. If you disqualify it for that reason, the title probably goes to the British Colossus the next year.
Obviously Zuse did develop conditional branching (such as in his language Plankalkül), but that was years later and by then it had already been done by others. To Zuse's credit, he probably would have gotten to it much faster had he not had a real problem with allied bombing destroying his work.
A major takeaway from this is that assigning titles like "first" and "inventor of computing" is quite silly. Computer science developed very, very quickly, with many of it key developments made by multiple people in parallel. We are largely familiar with what happened in the West for sociopolitical reasons, but similar work was done in Germany and the Soviet Union, often successfully. However, calling Zuse the inventor of computer science is as uncharitable to Turing as the reverse is to Zuse, both had their successes and failures.
When I give talks on computer history, I often like to say things like "Steve Jobs, inventor of the Computer" to elicit a laugh. The unfortunate thing is that sometimes people don't realize that it's a joke.