If this is correct, it was not a von Neuman machine originally, but it eventually became one, and at approximately the same time as the Manchester Baby.

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.

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.

The earlier relay computers were Turing complete.

For ENIAC it also does not make sense to claim that it was Turing complete. Such a claim can be made for a computer controlled by a program memory, where you have a defined instruction set, and the instruction set may be complete or not. If you may rewire arbitrarily the execution units, any computer is Turing complete.

The earlier ABC electronic computer was built for a special purpose, the solution of systems of linear algebraic equations, like ENIAC was built only for a special purpose, the computation of artillery tables.

By rewiring the ABC electronic computer you could have also computed anything, so you can say that it was Turing complete, if rewiring is allowed.

The only difference is that rewiring was simpler in ENIAC, because it had been planned to be easy, so there were special panels where you could alter the connections.

Neither ABC nor ENIAC had enough memory to be truly general-purpose, and by the end of the war it was recognized that this was the main limitation for extending the domain of applications, so the ENIAC team proposed ultrasonic delay lines as the solution for a big memory (inspired by the use of delay lines as an analog memory in radars), while von Neumann proposed the use of a cathode ray tube of the kind used in video cameras (iconoscope; this was implemented first in the Manchester computers).

Because ENIAC was not really designed as general-purpose, its designers originally did not think about high-capacity memories. On the other hand, John Vincent Atanasoff, the main designer of the ABC computer, has written a very insightful document about the requirements for memories in digital computers, years before ENIAC, where he analyzed all the known possibilities and where he invented the concept of DRAM, but implemented with discrete capacitors. Later, the proposal of von Neumann was also to use a DRAM, but to use a cheaper and more compact iconoscope CRT, instead of discrete capacitors.

While the ABC computer was not general-purpose as built, the document written by Atanasoff in 1940, “Computing Machine for the Solution of Large Systems of Linear Algebraic Equations”, demonstrated a much better understanding of the concept of a general-purpose electronic digital computer than the designers of ENIAC would demonstrate before the end of 1944 / beginning of 1945, when they realized that a bigger memory is needed to make a computer suitable for any other applications, i.e. for really making it "general purpose".