A [computer] program is a sequence of instructions executable by a machine (the computer).
> Would you argue that the Euclidean algorithm from two millenia prior was a program?
It's in the name: EA is an algorithm.
Btw, this human "compiler" job was for a long time considered to be inferior work and part of the reason why the first professional software developers in mid 20th century were largely women: it was considered clerical work to translate algorithms from paper into computer programs. The Computer Girls is a good article that describes these attitudes.
http://homes.soic.indiana.edu/nensmeng/files/ensmenger-gende...
The string "x=1" can both be a computer program, and something intended only for humans to read. The Euclidean Algorithm was written specifically for humans to understand, with no intent for them to ever be interpreted by a machine. The fact that someone at some point did implement it doesn't retroactively make it the first program. Lovelace's "Diagram" was also not something a machine could directly execute. But the key difference was the intention, she specifically intended that her instructions could be interpreted and executed by a machine.
Ada Lovelace was the first to realise that the analytical engine would perform arbitrary tasks and wrote programs for those arbitrary task, beyond computational operations. Of course Babbage who designed the hardware had some idea of what programs it could run and presented examples, but he did not have the forethought to go beyond as Lovelace is quoted in your articles, "an original understanding of where the power and potential of computers lay."
There has always been a controversy of how much of Lovelace's work is hers and how much is Babbage's in the Menabrea papers, and I don't think Babbage writing a few simple programs settles this controversy one way or another. Lovelace had unique and original insights that should not be downplayed.