OK, I'll give a serious answer.
My slant on this is that a player is deliberate if they often labor over a decision, even an easy decision. Such a player will typically have a slower than average preshot routine, too. Such a player, too often, uses up all the shot clock, and causes matches to last too long. Ralf Souquet is a good example, but his long-term excellence in the Mosconi Cup shows that he can keep it moving if he must.
A slow player, to me, is one having a slow decision process AND a preshot routine that takes forever. Such a player uses up most of the shot clock on every shot. Rising star and US Open runner-up Klenti Kaci is a good example, and the matches of such a player can take forever.
In the context of the original question posed, I'd say a deliberate player uses up most of the shot clock over 50% of the time and a slow player uses it up over 80% of the time.
Pro pool is trending toward slower and slower play, as the shot clock allotment has, in recent years, increased from 30 to 40 (sometimes even 45) seconds per shot. To quote the late, great Grady Matthews, "if you can't think of something to do in 30 seconds, maybe you should consider doing something else for a living." Amen.