My typical expectation is than when "subject" is used a context like this, it means the target of the certificate. If you think of a certificate as a cryptographically secured description of a thing (person, device, communication channel, etc), then the subject is the stuff related to that thing.
It's not the thing itself. For example, no one would say "the subject takes his SmartCard and authenticates his PIN". That would be the "user".
But it usually relates to the various data items related to that that thing. For example:
Usually, it's used to distinguish between the other players in the PKI world. Namely the "issuer" and the "root". The issuer is the CA that issued the cert (to the subject), and the root is the CA that is end point of all the trust in the heirarchy. The typical relationship is root--->issuer--->subject.