Yeah, this is a bit confusing, and the subject of repeated internal controversy. Most of the twentieth-century authors focused on the number of 1s, Σ(n), following Radó's original practice of treating σ(M) as the score of a machine M in the "Busy Beaver game". But when Aaronson re-popularized it in 2020 [0], he used BB(n) to denote the number of steps (which Radó called S(n)), and the bbchallenge project has been using this latter convention for publicity. Pascal Michel's website [1] has all the Σ(n) and S(n) bounds up to n = 7.

Personally, I think both functions have their strengths and weaknesses. Σ(n) is easier to calculate for machines that run too long to be simulated directly (e.g., Skelet #1 from the article) but leave a known pattern on the tape, and it also has historical priority. But S(n) has a simpler argument for being undecidable, since it provides a trivial filter for testing if a candidate machine cannot halt. Also, σ(M) is a bit weird in that it has no lower bound in terms of s(M), since an adversarial machine could do a colossal amount of work before wiping its tape at the end.

Regardless, past BB(3), there isn't any known size where the champion machines for Σ(n) and S(n) are different. (At least, the sets of champion machines aren't disjoint: Σ(5) = 4098 is shared by both the S(5) champion and another machine that runs a quarter as long.) The score of a machine is dominated by googological strength rather than technicalities in the definition.

[0] https://scottaaronson.blog/?p=4916

[1] https://bbchallenge.org/~pascal.michel/ha