There were programmers before that, e.g. for the IBM ASCC at Harvard, which was based on the ideas of Howard Aiken (inspired by Babbage).

Programming the IBM ASCC (a.k.a. Harvard Mark I) was much closer to the programming a modern computer in comparison with ENIAC, as it had an instruction set and programmers wrote a sequence of instructions on punched tapes. However even ASCC had some panels where it was possible to rewire some of the execution units to change their behavior, i.e. to change what some of the instructions from the instruction set did, but that was not the primary means for programming the computer. In ASCC the rewiring was akin to the microprogramming available in some later electronic computers, where you could change what some instructions did or you could add custom instructions.

Among the programmers of the IBM ASCC, Grace Hopper became later famous due to her contributions to the first high-level programming languages.

Therefore the profession of programmer has not started with ENIAC, even if the ENIAC programmers were among the first programmers.