The word "microprocessor" is typically used to mean "monolithic processor" a.k.a. "monolithic CPU", where "processor" a.k.a. "CPU" is a traditional term for a component of a general-purpose programmable computer, and "monolithic" means that the CPU is implemented in a single semiconductor chip.
Intel 4004 had nothing to do with general-purpose programmable computers. It was a chip used for making desktop calculators with fixed functions. Its architecture had nothing in common with the following microprocessors, from Intel and from other companies.
So I do not agree with Intel's claim. In my opinion, it is correct to consider that Intel 8008 from 1972 is the first microprocessor. Intel 8008, which implemented the architecture of the Datapoint 2200, was indeed a CPU for a general-purpose programmable computer.
The following mainline Intel processors, i.e. 8080, 8085, 8086 and so on, had ISA's derived from that of Intel 8008, and even the latest AMD Zen 4 and Intel Raptor Lake still include some derivatives of most of the 8008 instructions in their ISA's, for backward compatibility reasons.
The only reason why Intel 4004 has a place in the history of microprocessors is that this project has provided an opportunity for a team of Intel engineers to learn how to design complex digital integrated circuits in MOS technology.
Previously, Intel was a memory company. The logic design of memories was far simpler than the design of a CPU. Had Intel started directly with a general-purpose CPU, they would have needed much more time to discover how to design one.
After the experience with designing the Intel 4004, the Intel engineers have conceived a methodology of how to design complex logic circuits using their PMOS process, which they have then successfully used to design the Intel 8008.
If Intel 4004 would be considered as a processor, because it was also able to execute arithmetic operations, then it would not be clear where a line should be drawn. By that argument, why not also consider a monolithic ALU, like Texas Instruments 74181, as being a microprocessor.