Is there a word for that laugh you do, where you don't actually laugh, you just blow a bit of air out of your nose. Closest I can think of is "snort", but it doesn't seem right. "Snort" to me would be a louder noise, not just a quiet/subdued breathing out of a bit of air through your nose. I suspect based on my googling and seeing this question a lot, with no real answers, the answer will be no. But thought I'd ask anyway.
Snort is actually the right word, but you can describe it specifically, as in quietly (or lightly) snorted.
I don't know if there's a word for this laugh, but it's best not to get hung up on trivialities like this. Ask yourself how important it is that your readers know exactly how your character laughed. Just say 'Laughed half-heartedly' or 'Gave a short laugh'. Your readers will know what you mean.
I might say "laughed through their nose" but that could just give the impression of an annoying nasal laugh. I think it's one of those things that there's no specific word for and you just have to describe the action if you want to be sure about it getting across.
I've heard the phrase "nose laugh", and Googling finds it, but only in the Urban Dictionary, not a regular dictionary
I've been calling it a shuff (like a mix between snort/sniff and chuff), but urban dictionary said it's shnort
'A faint smile and a sharp breath of air through his nostrils was the only indication that he was amused by this. So, I guess what I'm saying is, in the absence of a single word you may need a sentence that covers it.
I'd be cautious about reaching too far for a word-- it risks sounding pretentious or even silly, and might distract the reader. And the same with using too many words to describe it. I like the idea of letting the reader modify it via the setting and the character development. I'd recommend picking up an old-fashioned print thesaurus, the kind with the numerical organization, and looking up "laugh" and "chuckle" and "snort" and seeing what words pop up. But I'm old.
I like this. Simple, expressive, descriptive. There's a time and a place for looking for that one perfect, if rather exotic, word, but the word for that time is not always. We've all done this, gotten stuck in the quest for that one perfect word, the word to perfectly encapsulate the thing. "Ah, look! The dictionary says this is called a snortle! Whodathunkit? Snortle!" If you make me look it up (and my vocabulary is very impressive), then the word didn't serve the purpose because I was forced to rely on the dictionary's dry definition of that word rather than your (possibly) more eloquent description.
My Oxford American Dictionary, 2d, defines snort as "an explosive sound." So I don't think one can snort silently, though one could perhaps stifle a snort. Or make a gesture of wry amusement. But a silent snort is as inherently contradictory as fake news.
I don't think it includes words. A horse can and does snort, but other than Mr. Ed none of them use words.
If you feel that may be too flippant, then how about this more action-packed variant. "His nose, a wind turbine of jocularity, silently blasted."