Was Lampsacus a Phocaean or Milesian colony?

Qualifiers

• This is one of those ancient geography questions that I like to go through, and hence enjoy solving -- but am never satisfied with the results (as it were).
• OP is probably more knowledgeable, hence I wonder if this perspective (not really an answer) was considered.
• More a leading consideration/factor, than an answer. But it could help OP resolve the question/issue of origins.

History Based on Evidence: First vs Secondhand Information

Strabo's Geography (Geographica), although full of place-names and locations, has a slight problem in general and it is a question of authenticity (does he really know these places). In other words, Geographica has lots of data, but it often comes without context, which is necessary and especially important in geographic work.

Here's the issue as it relates to OP's question: Pityusa or Pityussa is, I have read, Greek for "Pine Trees" or "Pine". During the time period under consideration, there could be more than one location with this name. The question of why did subsequent historians disregard or acknowledge Strabo (as OP stated: "yet this rejection of Strabo's account of Milesian origin is not shared by The Princeton Encyclopedia of Classical Sites (1976)") may very well rest on this fact, i.e. that there were several place-names of Pityusa/Pityussa.

Multiple Pityusa / Pityussa / Pitya

For the relevant book of Geographica, it is Book XIII - North Aegean. And from Bill Thayer's site, LacusCurtius (which is old enough to have a Wikipedia entry), there is a Pitya and there is another place, Lampsacus:

Pitya (fn.79) is in Pityus in the territory of Parium, lying below a pine-covered mountain;80 and it lies between Parium and Priapus in the direction of Linum, a place on the seashore, where are caught the Linusian snails, the best in the world

However, there is another Pityusa/Pityussa (the one OP is looking at, Lampsacus), later in this same chapter:

Lampsacus, also, is a city on the sea, a notable city with a good harbour, and still flourishing, like Abydus. It is about one hundred and seventy stadia distant from Abydus; and it was formerly called Pityussa, as also, it is said, was Chios. On the opposite shore of the Chersonesus is Callipolis, a small town. It is on the headland and runs far out towards Asia in the direction of the city of the Lampsaceni, so that the passage across to Asia from it is no more than forty stadia.

The entry for footnote 79 explains the issue and confusion that Geograhica sometimes generate:

f.n 79: According to the Scholiast on Apollonius Rhodius (1.933), cited by Leaf (Troy, p187), "Lampsacus was formerly called Pityeia, or, as others spell it, Pitya. Some say that Phrixus stored his treasure there and that the city was named after the treasure, for the Thracian word for treasure is 'pitye' " (but cf. the Greek word "pitys," "pine tree"). Strabo, however, places Pitya to the east of Parium, whereas Lampsacus lies to the west (see Leaf, l.c., pp185 ff.; and his Strabo on the Troad, p87). In § 18 (following) Strabo says that "Lampsacus was formerly called Pityussa.

Source: LacusCurtius

Hence, it requires very close reading of Geographica to find out which entry of Pitye/Pityusa/Pityussa is relevant to one's research.

I am not so sure this answers the question. In fact, I know it does not. But it does explain why we need to go beyond Strabo's place-names in Geographica to find the answer on questions of origin. Today, Lampsacus is Lâpseki, and according to Wikipeida (which isn't much), the answer is a colony founded by Phocaea.

(For place-names and origins, one needs to go beyond Geographica and Wikipedia. Hence, this is not good enough.)

SHORT ANSWER

Most modern historians lean towards Lampsacus being originally a Phocaean colony, this being based on an account of the local historigrapher Charon of Lampsacus in the surviving fragments of his Chronicles of Lampsacus, as well as other evidence linking Lampsacus to Phocaea.

Evidence for Milesian involvement is lacking, aside from Strabo's much later text, so your Option 2 seems most likely (as you surmised). Also worth noting is that, according to Charon's account, the Phocaean colony does not seem to have been started completely from scratch.

DETAILS

The earliest source for the founding of Lampsacus comes from a local historian, Charon, who pre-dates Thucydides and is believed to be roughly contemporaneous to Herodotus, or perhaps a little earlier: Charon was born either in the late 6th century or early 5th century BC. This makes him by far the earliest source we have for the founding of Lampsacus. Unfortunately, only fragments of his works survive, among which are some from the four-volume Hôroi Lam­psakēnôn (Chronicles of Lampsacus).

Although there are doubts over whether Charon really authored some of the works attributed to him in the Byzantine encyclopedia Suda, Hôroi Lam­psakēnôn is almost certainly by this historiographer from Lampsacus. Choron was cited by a number of ancient writers, including Thucydides and - more importantly for this question - Plutarch.

As already noted by the OP, Lampsacus was originally known as Pityussa (or a variant spelling). In De Mulierum Virtutibus (On the Bravery of Women or Virtues of Women), Plutarch writes:

There came from Phocaea twin brothers Phobus and Blepsus of the family of the Codridae, of whom Phobus was the first to throw himself into the sea from the Leucadian Rocks, as Charon of Lampsacus has recorded. Phobus, having influence and princely rank, sailed to Parium on some business of his own, and having become the friend and guest of Mandron, who was king of the Bebrycians who are called the Pityoessenians, he aided them by fighting on their side when they were being harassed by their neighbours. When Phobus took his departure Mandron expressed the utmost regard for him, and, in particular, promised to give him a part of their land and city if Phobus wished to come to Pityoessa with Phocaean colonists. So Phobus prevailed on his citizens and sent out his brother with the colonists.

(my emphasis)

Noted classical art historian and archaeologist John Boardman, in The Greeks Overseas is among those who, tentatively at least, attribute the colony to Phocaea, as do Lionel Pearson in Early Ionian Historians and M. H. Hansen and T. H. Nielsen in An Inventory of Archaic and Classical Poleis. The latter text provides further evidence for a Phocaean foundation:

That Phokaia was the metropolis is confirmed by the tradition that the citizens of Lampsakos called themselves brothers of the Massaliotai (cf. Massalia (no. 3), so a Phokaian colony; I.Lampsakos 4.26), and by the name of the month Heraion, attested both in Lampsakos (I.Lampsakos 8.5) and Phokaia (Samuel (1972) 125, 131).

Note: Massalia (modern-day Marseille) was a Phocaean colony.

Dennis R. Alley, in Charon of Lampsakos, despite referring to the "colonial foundation myth of Lampsakos", nonetheless cites the fragments of Charon as "an important portion of our surviving evidence" on the history of Lampsacus.

A. R. Burn in Greek Sea-Power, 776-540 B. C., and the 'Carian' Entry in the Eusebian Thalassocracy-List also supports a Phocaean foundation but asserts:

Milesians also supplanted the original Phocaean colonists of Lampsacus, but at what date we cannot tell (Str. xiii. 589; Charon of Lampsacus, frag. 6).

Burn seems to be making an assumption here, that the different accounts of Charon and Strabo meant that Milesians replaced the earlier Phocaeans. Neither author says this, and there is no evidence elsewhere either.

Excavations, the first of which was in 1996 (by which time the ancient city had long since been almost completely erased), have provided no evidence; almost all finds have been from later periods:

there have been no systematic excavations that would allow the reconstruction of the city’s full picture. Recent rescue excavations have brought to light sarcophagoi and significant tomb offerings and gifts, such as golden wreaths, diadems and a ring, all dating back to the second half of the 4th century BC.

Unfortunately, another early (c. 550 BC – c. 476 BC) historian and geographer Hecataeus of Miletus' works also only survive in fragments. He wrote stories on "the foundation of a town" but is not cited in any of the secondary sources on Lampsacus, and there is no mention of the colony in the collection of his fragments, Hecataei Milesii Fragmenta.

Strabo, writing at least 400 years after Charon, asserts that the Lampsaceni were Milesian colonists. Although this is not necessarily contradictory, evidence is lacking (as noted above). Nonetheless, it is possible that settlers from Miletus could have been welcomed at a later date to boost the strength of a colony which seems to have been surrounded by hostile neighbours at times. Eusebius of Caesarea, writing at least 800 years after the founding of colony, also attributed it to Miletus, his source probably being Strabo.

As to how Pityoessa or Pityussa came to be renamed Lampsacus, Plutarch relates the story of Mandron's daughter Lampsace and her role in saving the Phocaean colonists from the treachery of her own people in her father's absence:

...the daughter of Mandron, Lampsace, a young girl, learned of the plot beforehand, and tried first to dissuade her friends and relatives and to point out to them that they were undertaking to carry out a frightful and wicked deed in murdering men who were their benefactors and allies and now also their fellow-citizens. But when she could not prevail on them, she secretly told the Greeks what was afoot,....Lampsace died as the result of an illness, and they buried her within the city most magnificently, and called the city Lampsacus after her name.