ENIAC is notable because it was the first intentionally general purpose computer to be built.
[0] https://www.inf.fu-berlin.de/inst/ag-ki/rojas_home/documents...
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.
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.
True. Mauchly was a physics professor interested in meterology, and he knew that predicting the weather and calculating an artillery shell's flight are mathematically the same type of problem, which was important to get funding. In the fifties, Eniac was even used to calculate weather forecasts (see https://ams.confex.com/ams/2020Annual/webprogram/Manuscript/...). So these were just two related special problems, and it would be a stretch to interpret this as an intention to build a general-purpose computer. The latter had to wait until the sixties.
That's a pretty academic take. Neither Eckert, nor Mauchly, nor Zuse knew about Alan Turing’s 1936 paper when they designed their machines. The classification of ENIAC (and the Z3) as a "universal Turing machine" is entirely a retroactive reinterpretation by later computer scientists. John von Neumann knew the paper and was aware of its significance, but he only turned up in the ENIAC project when the design was complete. At this time, Eckert and Mauchly were already well aware of ENIAC's biggest flaw (the massive effort to reprogram the machine, and in fact they came up with the stored-program concept which von Neumann later formalized). ENIAC’s funding and primary justification were for the very specific purpose of calculating artillery firing tables for the military. The machine was built for this purpose, which included the feature which retroactively led to the mentioned classification.