If a student enrolls in a 4-year university full time, regardless of who is paying and regardless of whether he participates in sports, his ncaa "five to play four" clock is starting. It does not start when he "joins the team."
Typically a "gray shirt" candidate does not enroll in a 4-year university full time, so the individual's clock doesn't start. He or she may attend classes at a community college (and pay for it).
Here's one note "Basically a grayshirt is when an athlete delays his enrollment at his future college so that his eligibility clock will not start ticking until he arrives on campus during the second semester of the year."
A gray shirt candidate may not receive any benefits, such as meeting with team nutritionist, attending practices, working out in the athletic facility. A gray shirt usually has an agreement with the 4-year university that he will enroll the following year. This can be to time the athlete's entrance when it is most beneficial to the team and the athlete (too many in his position now, but a gap coming in the following year), or to allow recovery from a high school injury, or to get some academic experience if needed. "Gray shirt" is really just a fancy way of saying the team and the individual have mutually agreed to delay entrance to school and participation in team activities.