I'll come with my comments soon.

I shall start with some interesting fragments from Retorius and Valens dealing with the phases of the Moon

The new Moon signifies the glory and the power, dispositions of the king and of the authorities, the things related to public affairs of the state. For an individual it is relevant for his parents, partner, mysteries, and in generally everything that has a general and universal character. The same applies to the lord of the New Moon.
The rising of the Moon (15 degree from the Sun) , it is called light (phos), it significant for life, actions, the potentials, confirms the things from the New Moon.
From the first apparition it can be derived considerations about monthly universal movement, and the lord of this phase reveals the universal things.
Falcata (45 degree from the Sun) signifies the nutrition and the hopes that we have in life and also concerns the mother and the women in general.
The first quadrature it is significant for infirmities and illnesses, and every violent accident, but also for children, dignity, future benefits.
First Moon Gibbous (91-135 degrees) : properity, future promotions, departures, harmony of the couples.
The Full Moon: glory and obscurity, departures, violent things, about the things that are precipitated from the above and the things that are exalted from the base, the conformity of the feelings, aversions, adversaries, harmony of the genitors. is similar in significance with the seventh place from the horoscope (the text speaks about the occidental sign, so takes in consideration, in my opinion, the whole sign house system).
The diminish of light (arc of 60 degree from the Full Moon point - the text is corrupted and it has no mention about the degrees, but I took the reference from Paulus Alexandrinus Introduction) substance, diminishing of the activity, the things that start to diminish, sudden disgrace. Its virtue is similar to the eight house.
The second gibbous: expatriations, greate activities, prosperity, the same significance as the house of God (nine house).
The second quadrature: ancient matters (or maybe old?-my supposition), chronic sufferings, children.
Heliacal setting: ties, constrains, hidden things, accuses, disesteem.

One thing before the key of these phases in a natal chart interpretation:

don?t get angry for using greek ar latin, for me is the natural way, I feel that we have to come closer to the meaning of those languages, so just ignore them if you don?t like because I always use an equivalent English word. Another reason for doing this is because others might see another meaning in that term.

So again:

New Moon. Noum?ni, The Young Moon. Nea Sel?n, The Increasing Crescent. Hexag?nos: are associated with Mercury (lunar days: to the end of the eight day)

The Half Moon. Hemitomos: Venus (from the ninth day to the twelfth lunar day)

The Increasing Moon. Amphikurtos, Gibbous: Sun (from the 13 to the 14 lunar day)

The Full Moon. Pansel?nos (14-15 lunar day), and The Decreasing Moon. Amphikurtos (16-21lunar day): Mars

The second gibbous: Jupiter (22-25)

The Second Half-Moon. Hemitomos, The Decreasing Crescent. Menoeid?s, The Old Moon. En? Sel?n? (heliacal setting): Saturn (26-30)

How to use it in a natal chart. I?ll present in few words what is in the text in more detailed manner.

You must consider the lords of these phases, and the places of this phasis in the natal chart from the horoscope. The place of a certain phase signifies the start and the first stage, the lord of the phase the second epoch.
And now it comes the usual way of interpreting: good or bad places, aspects and so on.

These phases can be used also in an event chart, or a mundane chart or why not horary chart.

I think, and now comes my one speculations, that it would be better to analyze this phases from the part of fortune, considering the opinion of Ptolemy that the unchanged Part of fortune is in fact the lunar horoscope, but calculated in mundo. Anyway this is more an experimental field of my mind so you can take it as simple perverted idea from the canon.

The time is killing me so I will stop here for the moment

the sources of the fragments:
Rhetorii Aegyptii capitula selecta, CCAG VII/4, Bruxelles 1921, pag. 203-205.
Vettius Valens's Anhology, II, 5 Kroll. II, 36 Pingree

Sasha this is wondeful - thank you. And I completely support the way you have introduced the terminology here.

I would only like to add some bits to Sasha's awesome work.

Valens has 11 phases of the Moon in chapter 36 of book II (Pingree edition) in this way (with the names mentioned in his text):
1. Conjunction (sunodos)
2. Rising (anatol?)
3. Crescent (phasis m?noeid?s, separating Sun by 45 degrees)
4. Half (dichotomos, up to 90 degrees)
5. Double-Convex (amphikurtos, up to 135 degrees)
6. Whole Moon (pansel?nos, up to 180 degrees)
8. Second Double-Convex (deutera amphikurtos, up to 225 degrees)
9. Second Half (deutera dichotomos, up to 270 degrees)
10. Second Crescent (up to 315 degrees)
11. Setting (dusis, up to 360 degrees)
Phase 7, waning, is only briefly mentioned at the end of the paragraph.
As you see, neither phase 2 nor phase 7 is clearly defined which may be the result of the deficiency in this part of text, as this chapter only survived in codex Oxoniensis Seldenianus 22 which was written around 1520.

Fortunately, Valens' discussion was utilized by Rhetorius in the early 7th century. It can be found in Parisinus graecus 2425 as chapter 79 of the so-called Epitome III, which is that we can label as book V of Rhetorius. (The text itself makes Appendix VI in Pingree's edition.) In this passage phase 2 is constituted as 15 degrees from the sun, but even more interesting is that phase 7, which is here labeled as m?noeid?s again, is said to be from Full Moon until 165 degrees. Rhetorius seems to count in reverse order from this phase, and this may mean 195 degrees. However, we cannot be certain, since phases 8 to 10 slipped out of this manuscript.
Derived epitomae might help us but the corresponding chapter 23 of Epitome IIIb in codex Monacensis graecus 287 and Mutinensis graecus 85 are badly corrupt (CCAG 7: 116-7), in others it is missing or too short to clarify anything.
Fortunately, as chapter 10 of the so-called Epitome IV in codex Parisinus graecus 2506 and others manuscripts there is a well-preserved abbreviation of the Rhetorius passage. (CCAG 8.4: 203 footnotes) There are degrees allotted to phases which are 1, 15, 45, 90, 135, 180, 135, 90, 45, 15, 1 respectively.

It could be continued with investigation of Paulus who clearly derives from a slightly different tradition from Valens; of the Olympiodorus commentary of his; of a possible other fragment of Rhetorius from codex Berolinensis graecus 173 (CCAG 7: 222-3) which elaborates this very issue in some length; and of course later authors as well. But I did not want to go too far.

Thanks Osthanes for your sharp observations. I like your rigurosity.