What, specifically, are you referring to? The Bernoulli numbers algorithm was contained in Note G, which was an appendix she wrote in collaboration with Babbage.

I wouldn’t call it an urban legend if a Guardian article says “Her notes include what is recognised as the first algorithm intended to be processed by a machine, while she also speculated on its future ability to create graphics and complex music.”
https://www.theguardian.com/technology/2012/dec/10/ada-lovelace-honoured-google-doodle

(Referenced from the wikipedia article says “She was the first to recognise that the machine had applications beyond pure calculation, and to have published the first algorithm intended to be carried out by such a machine.”
https://en.wikipedia.org/wiki/Ada_Lovelace )

Babbage had conceived of simple programs and Menebrea’s paper includes some programs so I can see why there is debate about calling her program the first one. Seems like it was the first most complicated one though.

Don’t reduce their technical contributions to literary contributions.