There is some controversy over what to call the various “top-X”/“jungle primary” electoral systems now in vogue. The National Conference of State Legislatures now offers the term “multi-party primary” with the following definition:
A small but growing number of states hold a single primary in which all candidates, regardless of party, are listed on a single ballot. States vary in the number of candidates who advance out of this primary to the general election…
Elections for Nebraska’s unicameral, nonpartisan legislature closely mirror this process except ballots do not identify the candidates’ party.
I prefer the term “nonpartisan two-round system” (NPTRS) for two reasons. First, I take my definition of “partisan election” from literature in which some formal process governs a candidate’s use of the party label on a ballot, i.e., there is an actual endorsement. Second, some of that literature is in comparative politics, and since the goal is to import a “comparative” electoral system, it may help to bring with it the terminology used to analyze the working of such systems.
Another way to make the second point is to note that these reforms at least pretend to favor multi-party politics (see the name given above to the first-round election). But there are no parties, formally speaking, and that may matter for their long-term operation.
If we need a term for only the first round, I prefer winnowing or preliminary election. This is the term used in some cities to describe an election whose purpose is to reduce the number of candidates proceeding to the next round (typically but not necessarily a general election).
Controversy about using the word “nonpartisan” boils down to the fact that party labels do appear on some states’ ballots (as described above). However, on closer inspection, these labels are just indications of party affiliation, and we all get to choose the one we want when we register to vote. It’s not a party endorsement arrived at either collectively, or via some collectively legitimated process written down in either public law or party bylaws.
I might be okay with “jungle primary” simply because many instantly know what it means. “Top X” or “top C” also works and may carry less baggage, although it is not widely used.