As mentioned in the question, Whitewright (2011) has a section titled, "Literary references to close-hauled sailing" from ancient sources which might help give a lower-bound for some of the terms.
Aristotle in the 5th c. BC wrote:
Why is it that, when the wind is unfavourable and they wish to run
before it, they reef the sail in the direction of the helmsman, and
slacken the part of the sheet towards the bows? Is it because the
rudder cannot act against the wind when it is stormy, but can when the
wind is slight and so they shorten sail? In this way the wind carries
the ship forward, but the rudder turns it into the wind, acting
against the sea as a lever. At the same time the sailors fight against
the wind; for they lean over in the opposite direction.
Note the translation uses the word "run" here (but Whitewright interprets it as describing a close-hauled course).
Pliny (circa AD 77) wrote:
Vessels by means of slacking the sheets can sail in contrary direction
with the same winds, so that collisions occur, usually at night,
between ships on opposite tacks.
Note the translation uses the word "tack" there.
Achilles Tatius (1st-2nd c. AD) wrote:
On the third day of our voyage, the perfect calm we had hitherto
experienced was suddenly overcast by dark clouds and the daylight
disappeared, a wind blew upwards from the sea full in the ship’s face,
and the helmsman bade the sailyard be slewed round. The sailors
hastened to effect this, bunching up half the sail upon the yard by
main force, for the increasing violence of the gusts obstructed their
efforts; for the rest, they kept enough of the full spread [of the
sail] to make the wind help them to tack. As a result of this the ship
lay on her side, one bulwark raised upward into the air and the deck a
steep slope, so that most of us thought that she must heel over when
the next gale struck us. We transferred ourselves therefore to that
part of the boat which was highest out of the water... the wind
suddenly shifted to the other side so that the ship was almost sent
under water, and instantly that part of the boat which had been down
in the waves was now violently thrown up... all changed their station,
running, with shouts and cries, to the position in which they had been
before they moved; and the same thing happened a third and a fourth,
nay, many times, we thus imitated the motion of the ship.
This translation also uses the word "tack".
Whitewright interprets all of these quotes jointly:
All these passages, in different ways, recount the experience and
practice of sailing a Mediterranean square-sail vessel on a
In summary, judging from these translated quotes from up to the 2nd c. AD, there seem to be phrases that can be translated as "run" or "tack", but no idioms being used analogous to "reaching" or "close-hauled" -- even when that's how the author Whitewright interprets each of the described scenarios.
Here I'm digging into some of the original text because I fear the presence of "run" and "tack" in the quotes may be translational flourishes.
Here's Pliny in the original Latin (Natural History, Vol. I, Sec. 2.48):
isdem autem ventis in contrarium navigatur prolatis pedibus, ut noctu
plerumque adversa vela concurrant
An alternative translation is (Bostock/Riley): "We are able to sail in opposite directions by means of the same wind, if we have the sails properly set; hence it frequently happens that, in the night, vessels going in different directions run against each other."
My amateur literal translation of the latter phrase would have it, "so at night frequently opposing sails collide". That is, it looks like the word "adversa" is indicating the adverse/opposing/directly-facing situation, and the "tack" is an addition by the one translator.
Here's Aristotle (or someone like him) in the original Greek (Mechanical Problems, 851b.7):
Διὰ τί, ὅταν ἐξ οὐρίας βούλωνται διαδραμεῖν μὴ οὐρίου τοῦ πνεύματος
ὄντος, τὸ μὲν πρὸς τὸν κυβερνήτην τοῦ ἱστίου μέρος στέλλονται, τὸ δὲ
πρὸς τὴν πρῷραν ποδιαῖον ποιησάμενοι ἐφιᾶσιν
Unfortunately, I don't know Greek, and online translation tools are failing miserably in this case. The best I can tell, the phrase "μὴ οὐρίου" (mí ouríou) says something like, "wish-not to wind-submit". So again it looks like the use of "run" there is a modern flourish by the translator, and possibly just totally contrary to how sailors use the word. (In ancient Greek, I think "run" would be "τρέξιμο", and there's no form of that word in the quoted passage.)