She was definitely the first person to realize the potential that Babbage's Analytic Engine had, particularly outside of just calculating things. Babbage's design, if I remember correctly, was basically an improvement on a design he initially made to calculate trajectories for artillery teams, and most of his thoughts on what to use his computer for were "calculate (thing)". Lovelace wrote on how the computer could be programmed to solve more complex problems.
I think that it would be a mistake to say that computer science has truly been around for 160 years, though. A few people (there was also an Italian who was interested in Babbage's work, although I'm not sure what his contributions ended up being, if any) does not a field make, and the fact that any progress in it was more or less put on hold until the early 20th century (when mathematicians started working on what you could calculate or construct in a finite number of steps), and you didn't get (untyped) lambda calculus and turing machines until 1936, which is probably the best place to truly start the idea of computer science as a field. (And since you got early devices that were sort of primitive mechanical computers in the late 30s, early 40s as part of the whole Bletchley Park cryptography work by the Brits.)
A very large debate around the turn of the century was if mathematics that you couldn't specifically construct in a finite (or countable) manner were, which became particularly heated after Cantor's set theory work (showing that the real numbers were uncountable) and then things like Russel's paradox (showing contradictions in Cantor's naive set theory if you allowed sets that contained themselves). I'd argue (without firm, researched proof that it was definitely the intent and case) that the spirit of early computer science (lambda calculus, turing machines, etc.), which was concerned with what you could and could not compute with a finite algorithm, came in spirit from those sorts of debates. (See finitism, intuitionism, constructivism, etc. for parts of this debate; traces of it remain in modern day mathematics with some people's concerns about if the Axiom of Choice is a valid or reasonable axiom to have)
I think you're mixing up a couple stories there, but it doesn't really matter that much. Just to fill in the details (and because the alternative is for me to start responding to work emails which I just don't feel like doing this morning): Babbage was concerned about the quality of mathmatical table books, which were used for calculations before we had modern calculators and slide rules weren't mainstream. The figures in them were computed by hand, and some were so bad there was an error on every page, which made them useless for scientific work and dangerous for industrial work. The Difference Engine was designed to print out pages of figures using a little printer attachment.
ENIAC was ininitially put to work calculating artillery trajectories, once the programmers figured out the bug where the shell kept going after it hit the ground (oops!).
