I do not think that it is right at all to say "intentionally general purpose computer".

ENIAC was built for a special purpose, the computation of artillery tables.

It was a bespoke computer built for a single customer: the United States Army's Ballistic Research Laboratory.

This is why it has been designed as the digital electronic equivalent of the analog mechanical computers that were previously used by the Army and why it does not resemble at all what is now meant by "general-purpose computer".

The computers of Aiken and Zuse were really intentionally general-purpose, their designers did not have in mind any specific computation, which is why they were controlled by a program memory, not by a wiring diagram.

What you claim about Z3 being general purpose by accident does not refer to the intention of its designer, but only to the fact that its instruction set was actually powerful enough by accident, because at that early time it was not understood which kinds of instructions are necessary for completeness.

All the claims made now about ENIAC being general-purpose are retroactive. Only after the war ended and the concept of a digital computer became well understood, the ENIAC was repurposed to also do other tasks than originally planned.

The first truly general-purpose electronic digital computers that were intentionally designed to be so were those designed based on the von Neumann report.

Before the completion of the first of those, there were general-purpose hybrid electronic-electromechanical digital computers, IBM SSEC being the most important of them, which solved a lot of scientific and technical problems, before electronic computers became available.

A counter argument is that Mauchly was actually interesting in using computers for weather modeling and I’m sure that influenced the design of ENIAC. He could only get ENIAC funded if it was valuable to the war effort. I’ve read quite a lot about that machine and I’m not aware of any architectural features that were specific to ballistics calculations. This is unlike the British Colossus, another early computer, which was specifically designed for code breaking and wasn’t general purpose.

As for the objection that it wasn’t stored program, I was interested to learn that it was converted to stored program operation after only two years or so of operation, using the constant table switches as the program store. But the Manchester Baby, which used the same memory for code and data was more significant in the history of stored program machines.

On the general question of “first computer”, I think the answer is whatever machine you want it to be if you heap enough conditional adjectives on it.