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About the only one I can stomach is Captain, and that with reservations.

1) It shouldn't strongly imply military connections (not all warlords have military backgrounds)

2) It shouldn't imply level of achievement

3) It shouldn't have a flavor that conflicts with any character concept within the class (I think many nobles without heavy military training should nonetheless be warlords)

Captain minimally works, since it has very broad usage (including non-military) and can encompass any level of achievement. Its flavor doesn't work well for some builds, but it doesn't totally clash with many.

Most of the suggestions fail on at least 2 of those criteria. I'm glad that R&D are hard at work on this.

I think that Harbinger is fine in its technical military definition, but the term has a "bad omen" connotation that keeps people from seeing it as a good fit. Maybe that's just me, but I usually think of harbingers in the abstract as being harbingers of something scary. I do think that it's another name that's good enough that if they just picked it, but it sounds a bit sinister to me in a vacuum. (Maybe I'm totally wrong about the connotation, but I think that it's there.)

Just to note, I did say that 'warlord' was not a formal position.

"The 'Lord' element implies a very high, if informal, rank, that being, the leader of an army."

I suppose anyone could call themself a warlord, even someone on their own ("I command the legions of my imagination!!!"), but someone doing so with only a handful of (non-imaginary) followers wouldn't likely be able to wage much of a 'war,' at least, not as it has been generally defined for a few thousand years now. So, if the 'lord' argument isn't working for you, you might consider this 'war' argument then. War implies scale, just as lord implies hierarchy, both of which to an order of magnitude that typically does (and I would contend, should) elude a 1st level 'warlord.' 

Now... I suppose you could argue that one could wage an aysymmetrical war, like an assassin's war, an information war, a narco-war, a 'terroristic' war and so on, with a small group of people, but these are all going pretty far afield of what a nominal warlord in DnD is and does.  

Personally, I also kind of like the archaic Athenian 'Strategos,' even though it's wildly innappropriate from both an historical and performative perspective. But the thing I keep coming back to with Commander, is that most of these other names COMMAND. 

The exceptions to this are: tactician, champion, harbinger, herald, vanguard, and possibly knight (depending on the incarnation). 

Tactician I'd be fine with. 
Champion is too focal, too front-line-associated, and not necessarily having much to do with planning. It's also a little too broad in the respect that any other class could easily be a 'Champion.'
Harbinger is awful imho. Signals to me the 'front of the front' line and is more of a messenger than a troop, and even less of a leader (you would never send your leader to give a portentous message to your enemies cuz they'd kill them. That's why you send a messenger/harbinger in the first place, they are of little overall consequence.
Same with Herald, though it is marginally better than Harbinger, since they could blast a horn or something while with their own troops.
Vanguard has the same problems as Champion; it doesn't imply any kind of leadership universally, beyond literally being  at the head of a formation, which is again, too front-line.
Knights are terrible, there is just way too much historico-semantic baggage. Are they blessed by god? Do they take vows of fealty or chastity or martyrdom? are they mounted combatants? do they get a troop of yeomen and a squire? or are they more like Sir Ian McKellan? or Sir Elton John? ... it's just far to squishy a term.