If the Vice President finishes out a term, does that count as a term for them?

It depends on the remaining time in the term they are finishing. A vice president that finishes a term someone else was elected to cannot be elected to their own term in office more than once if the term the finish has more than two years to go.

Examples:

LBJ had served less than a year of JFKs unfinished term when he was elected in 1964. He could have run in 1968 but elected not to.

Ford succeeded Nixon about 18 months in and had served more than two years when he ran in 1976. Had he won in 1976, he could not have run for a second term in 1980.

Busterwasmycat is right. The Constitution is very specific about the 2 year rule. Anything over 2 years counts as a full term. But Biden is in very good health. 

It depends on how long.  Go see the 22nd amendment.  A person who occupies the presidency without being elected to it, can hold that office for up to two years without it counting as a term under the term limits rule.  More than two years, and it counts as a term, and the person would only be eligible for one more term.  Less than two years, and the person would be eligible for two more terms.

No, it doesn't. If a Vice President takes over the remainder of a President's term because of death or incapacity, that Vice President can still serve two complete terms of their own. Even if the President dies the day after being inaugurated, the Vice President can still be elected to and serve two complete terms as President in addition to serving out the remainder of the prior President's term.

Two or more years counts as a term.