Static vs ephemeral diffie-hellman
Are &quot;private keys&quot; in the context of diffie-hellman refer to the private $a$ and $b$ that Alice and bob privately select respectively?

Yes, correct. Similarly, $A=g^a\bmod p$ and $B=g^b\bmod p$ are also called the &quot;public keys&quot;.

What is considered an exchange? A session of information exchanging between to parties?

An exchange is an execution of the Diffie-Hellman protocol. This happens whenever either a party feels like it or a party has forgotten the resulting shared secret from the last execution. In the context of the internet this usually means that if you connect to, say, Stack Exchange today it will run a Diffie-Hellman exchange. Then when you shutdown your computer it will &quot;forget&quot; the resulting shared secret $K$ and tomorrow when you connect to Stack Exchange again, it will have to run another Diffie-Hellman exchange.

If so, does static DH refer to exchanges between the same two parties?

In a static exchange usually both parties always reuse their private keys, which implies that if they re-run the Diffie-Hellman exchange more than once with each other they always get the same shared secret $K$.
Indeed getting a shared secret the same shared secret between Alice and Bob and Alice and Charlie has a negligible probability if everyone involved picked their private keys properly and the protocol was followed.