See this page on KosherShaver.org for what looks like a pretty thorough background on all of the issues.
Rav Moshe Feinstein understood the aforementioned sugyah in accordance with this latter approach. According to this approach, when the Gemara rules that an item which both destroys and shaves is prohibited, and defined this item as being a razor, it did not merely mention a razor as an example of an item that achieves both of these requirements, but mentioned it in order to define which instrument is forbidden by the Torah to use. The Gemara explains that when the Torah forbade destroying and shaving one's beard, the Torah was referring to a razor which achieves both of these functions, and not to other instruments which are categorically permissible. It follows that even if one manipulates scissors to enable it to achieve השחתה, it is still permitted, since the Torah did not forbid the end result, but forbade the use of a specific instrument to achieve this result. What categorizes an item as scissors, which is permitted, or as a razor, which is forbidden, is the fact that a razor cuts using one blade, whereas scissors use two blades to cut.
Based on this thesis, Rav Moshe permitted the use of shavers since they function as scissors, utilizing two blades to cut. The inner blade does not cut by itself, and must be assisted by the outer screen of the shaver. The screen traps a hair within it, and as the inner blade approaches it, the hair rubs along the side of the screen and they both cut the hair simultaneously.28
Although this was Rav Moshe’s primary heter to permit the use of shavers, he had an additional approach that followed a similar line of reasoning. He maintained that the only type of instrument that is prohibited, is one that most people use when they wish to achieve a close shave; all other instruments are permitted. Since most non-Jews use razors when they desire a close shave, all other instruments are therefore permitted. However, if for arguments sake, the general custom would change and people would only use shavers even if they desired a very close cut, razors would be permitted and shavers would be forbidden.29
Footnote 28: Thus I heard in the form of "reliable testimony" (עדות נאמנה) from Rav Belsky, Rav Dovid Feinstein and Rav Reuven Feinstein
Footnote 29: Thus I head from Rav Reuven Feinstein
Later, with regard to lift-and-cut:
It is important to mention that Rav Moshe Feinstein was very skeptical about permitting the use of a lift and cut shaver when it was first shown to him in his later years. One should bear in mind that the widespread heter to use electric shavers is primarily because Rav Moshe, being the posek hador of the past generation, permitted their use. Most poskim in Eretz Yisroel on the other hand, forbade their use. Consequently, it is questionable how a person who relies on Rav Moshe’s shitah with regard to shaving, can go ahead and use a lift and cut shaver which Rav Moshe himself was skeptical about.31
Footnote 31: This is what I heard from Rav Belsky, Rav Dovid Feinstein and Rav Reuven Feinstein. See Halichot Shelomo page 11 who gives another rationale to permit shaving...
The site says that it is put out by Halacha Berurah and lists authors at the top of the page.
It would seem that there is no written heter in Igrot Moshe for shaving. However, despite this, Rav Moshe's (and Rav Heinkin's) position on this seems to be pretty well known (the author quotes two of his sons directly).
Also see this page under the heading UPDATE: FEBRUARY 2011 / ADAR I 5771 where the author provides what he claims to be up to date psakim by Rav Dovid Feinstein, Rav Belski and Rav Heineman.