What is considered part of the beard, and what hair on the face is not, and therefore permissible to shave or trim?
The majority of classical Arabic linguistic scholars consider the lihya (from lahi) to be the hair on the chin, plus on the entire jawbone. A minority say chin plus front teeth only. The vast majority also include hair on the cheekbone as well, while saying that the hair below the eyes and on the throat is not part of the lihya. There is further a difference of opinion on the hair below the chin and above the throat.
Reference: The above is taken from a class on fiqh of clothing taught by Dr. Yasir Qadhi. Other answers directly referencing books of fiqh are preferred - I will delete my answer when those show up.
Majority of the Scholars understand beard from the linguistic definition of "lihyah" which means the hair that comes from the chin and cheeks.
Ibn Manzoor said quoting from Ibn Sayyidih:
Lihyah (beard) includes the hair that grows on the cheeks and the chin. Lisaan al-‘Arab, 15/243
Similarly it also says in Fath al-Baari, p. 35; vol. 10, al-Salafiyyah edition:
It is a name for that which grows on the cheeks and chin.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) was asked about what is meant by the beard. He replied:
The extent of the beard is from the cheekbones, level with the channel of the ears, until the bottom of the face. It includes the hair that grows on the cheeks.
It says in al-Qaamoos al-Muheet (p. 387): Lihyah (beard) is the hair of the cheeks and chin.
Based on this, the one who says that the hair that grows on the cheeks is not part of the beard must bring proof of that.
Majmoo’ Fataawa Ibn ‘Uthaymeen, 11/question no. 49
He (may Allaah have mercy on him) was also asked: Are the cheeks part of the beard?
Yes, the cheeks are part of the beard, because this is what is implied by the language in which sharee’ah came. Allaah says (interpretation of the meaning):
“Verily, We have sent it down as an Arabic Qur’aan in order that you may understand”[Yoosuf 12:2]
“He it is Who sent among the unlettered ones a Messenger (Muhammad) from among themselves, reciting to them His Verses, purifying them (from the filth of disbelief and polytheism), and teaching them the Book (this Qur’aan, Islamic laws and Islamic jurisprudence) and Al‑Hikmah (As‑Sunnah)” [al-Jumu’ah 62:2]
Hence it is known that the meanings brought in the Qur’aan and Sunnah are the meanings indicated by the Arabic language, but if there is a particular shar’i meaning it should be interpreted in that manner. For example, salaah in Arabic means supplication (as in du’aa’), but in shar’i terminology it refers to the specific, well known act of worship (prayer), so when it is mentioned in the Qur’aan and Sunnah, it is to be understood in the shar’i sense, unless there is any indication to the contrary.
As for the hair that grows on the neck, it is not part of the beard. Imam Ahmad (may Allaah have mercy on him) stated that there is nothing wrong with removing hair that is beneath the pharynx, as it says in al-Insaaf (1/250), because it is not part of the beard.
Shaykh Muhammad al-Safareeni said:
The reliable point of view, as it says in al-Iqnaa’ and elsewhere, is that it is not makrooh to remove hair that is beneath the pharynx.
Ghidha’ al-Albaab Sharh Manzoomat al-Adaab, 1/433
Source and further reading: Definition of the beard and ruling on shaving the hair that grows on the neck, What is the hair that it is permissible to remove and the hair that it is not permissible to remove?