This doesn't work, and I have no idea how to ask for if an edge exists in the phabricator_maniphest.edge table with a type of 43 and references this task.

And this is just copy paste fail.

Add a JOIN to the edge table here, e.g. JOIN edge column ON edge.src = task.phid AND edge.type = 43

I think things aren't sticking because these are Phids but should be PHIDs.

e.g., this is "type a user, project or package name".

I modified the PhabricatorProjectColumn->getDisplayName() to show "ColumnName (on board Board)" so that it showed up correctly in the tokenizer. Otherwise, it's impossible to tell which board the 20 "Backlog"s are coming from. However, now that shows up everywhere else and is horribly ugly. I'm not sure how to make it show up that way only in the tokenizer.

Ugliness:

Without this change, this is what you get when searching:

I added a type hint, because it looked strange to have this one be empty when most the other fields had one. Should I change that?

{F168174}

Without the type hint:

{F168178}

Additionally, because I've made it so the type hinting works on the column OR the project name, I think it's beneficial to inform the user that they can type either.

There's no great way to fix the name thing in general. You can hard-code whatever you want here, which will fix the typeahead dropdown menu. However, it won't affect tokens that are pre-filled into the input on page load, since those will use the names of their handles. There's no reasonable way to avoid this right now. You could go find that code (AphrontTokenizerTemplateView) and do something hacky there, maybe (in render(), where renderToken() is called).