This
with "--primary" specified, it will count each multi-mapping read once
is not entirely true (at least for featureCounts v2.0.0). featureCounts recognizes multimapped reads using the NH:i tag. If you use --primary without -M featureCounts will count only uniquely mapped reads (NH:i:1) and completely ignore multimapped reads regardless on the SAM primary tag. Using --primary in this scenario doesn't have any effect on counting whatsoever.
In case the aligner doesn't output the NH:i flag, featureCounts doesn't recognize unique/multimapped and will count all the alignments. In this case, using --primary will help you to count each read only once.
If you want to count both uniquely and multimapped reads (and your aligner outputs the NH:i flag) but each of the reads only once you can use --primary -M.