If the city uses partisan elections or the nonpartisan primary and election method, is the first city "election" the primary, or is it the general election?

That's not absolutely clear, but the North Carolina election law, in G.S. 163-291 and 163-294, does distinguish between the "primary" and the "election," using the word "election" to refer only to the general election. That usage suggests that the 90-day deadline applies to the general election and not to the preceding primaries.