What is the hair of the beard? Does it include the hair on the neck (just above the throat) or can I shave that and just let it grow on my cheeks, chin, and lower jaw as some scholars permit?

There are two aspects to your query; a) Growing of a Waajib (compulsory) beard, and b) Growing of a Sunnah beard.

In principle, whatever grows on the jaw bone is considered as beard. The Hair on the neck and cheek is not considered as part of the beard. However, according to all four Imaams of Fiqh, it is necessary to have a fist long beard all round. In view of this, practically, a person having a fist long beard only on the jaw bone shall be odd and surprising.

Therefore, the hair on the neck that is growing towards the throat may be cut but the hair that is growing towards the beard should not be cut.

and Allah Ta'ala Knows Best

Mufti Ebrahim Desai