The first true (civilian) microprocessor was Intel 8008, in 1972.
Intel 8008 was a monolithic implementation, i.e. in a single PMOS integrated circuit, of the processor of Datapoint 2200, therefore it deserves the name "microprocessor".
The processor of Datapoint 2200 had an ugly architecture, but there is no doubt that it was a general-purpose CPU and traces of its ISA remain present in the latest Intel and AMD CPUs.
On the other hand, the set of chips that included Intel 4004 was not intended for the implementation of a general-purpose computer, but it was intended just for the implementation of a classic desktop calculator, not even a programmable desktop calculator.
This is the reason for the many quirks of Intel 4004, e.g. the lack of instructions for the logic operations, and many others that have increased the amount of work required for implementing a MIPS emulator suitable for running Linux.
Even if Intel 4004 was intended for a restricted application, after Intel has offered to sell it to anyone, there have been many who have succeeded to use it in various creative ways for implementing microcontrollers for the automation of diverse industrial processes, saving some money or some space over a TTL implementation.
In the early days of the electronics industry it was very normal to find ways to use integrated circuits for purposes very different from those for which the circuits had been designed. Such applications do not make Intel 4004 a true microcontroller or microprocessor. Very soon many other companies, and later also Intel, have begun to produce true microcontrollers, designed for this purpose, either 4-bit or 8-bit MCUs, then Intel 4004 has no longer been used for new designs.