It seems people call different things "full nodes".

The wiki here defines it simply as a node that has personally verified the whole blockchain, and that continuously does so for new blocks and transactions. I suppose that also implies relaying only valid transactions to its peers. According to the wiki, a full node does not have to be an "archival node" which stores the whole blockchain and accepts incoming connections.

By that definition, miners are necessarily full nodes, but I'm not sure if they must also be archival nodes.

Here's a 2014 thread where someone asks about full node rewarding.