When a female takes a male back to her room or goes back to his room, but does not want to sleep with him and instead calms him down or deflects his advances by giving him a defensive handjob. It is only a defensive handjob however if the evening's activities stop there, for instance intercourse immediately changes it from a defensive handjob to the normal foreplay variety of handjob.
When a girl just plays with your dick a little bit. Does not jerk you off to completion. This move originated at California State University, Stanislaus, and is usually performed before a blowjob.
"What's a quasi handjob?" -Danny Mullen
"She just, like, played with my dick a little" - Leo Dottavio
"Leo was in a Home Depot parking lot and asked a broad with low self esteem for a quasi handjob before she blew him"