Attention English teachers and grammar Nazis: what is the correct punctuation?

The structure of the first one seems fine to me. It is an introductory clause followed by three main clauses separated by commas. Each main cause has a Subject and Verb. Structurally it would be like the following: "At the spa, Judy got a manicure, Pam got a haircut, and I read a magazine." Technically, you could separate the main clauses by semicolons, like you did in the second one, but that seems odd to me.

The alternative structure would be a main clause followed by a colon and a list. "The girls and I went to the spa: Judy got a manicure, Pam got a haircut, and I read a magazine."

Or, for your sentence: "the variables in the equation above are as follows: a is one, b is two, and c is three."

Semicolons can be used to separate items in a list but generally are only used when it is necessary for clarity. E.g. if the listed items are long or have internal commas. Say for instance you had to say something like this: "a, b, and f are one; c, d, and e are two; g is three."

I am not an English teacher or grammar nazi, so please don't take this as gospel.