Ada Lovelace is usually credited with writing the worlds first computer program for Charles Babbages Analytical Engine. The program in question is her legendary 'Note G', published in 1842, for calculating the series of Bernoulli numbers.

Some time ago, I stumbled over this article written by a professor for computer science at the Free University Berlin:

https://www.heise.de/tp/features/Die-ersten-Computerprogramme-der-Welt-3315325.html?seite=all

Unfortunately the article is in german, so I will translate the central passage relevant for my question:

"In the Science Museum London one can find 27 programs in the 'Babbage Archive' which the scholar wrote between 1837 and 1840 for the Analytical Engine. They are very interesting because they reveal so far unknown details about processor and memory architecture [...] On the same time, this source code is the very first one that was ever written for a computer - it would be unneccessary to mention that Babbage is therefore the worlds first computer programmer, wouldn't indefensible urban legends claiming something else be circulating in the Internet."

It should be quite obvious that 'Ada Lovelace as worlds first computer programmer' is what the author means with 'indefensible urban legend'.

What is known about Charles Babbages programs mentioned here, some of those look to be written 5 years before Lovelaces publication? Who really wrote the first program for the Analytical Engine, Babbage or Lovelace?