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.
I think deliberate play is walking around the table before every shot to gain a clear picture of the shot you are on and the next 2 shots. Slow play is when someone stands and stares at the table for a full 60 seconds without moving a muscle, or getting into shooting position then standing back up to re-evaluate the same shot several times. When shooting an opponent you can get usually get a pretty quick idea which of the 2 you have.
If he has the nickname Cobweb, he's a slow player...
If you take a bathroom break and he's still racking when you come back....he's a slow player.
He has a shot on the 1, so you duck out for a quick smoke....He's looking at the 4 ball when you come back and doesn'r realize you left.
If he spends more than 30 seconds chalking and can't look at the table simultaneously...he's slow.
He tells you his wife was pregnant for over a year....he's a slow player.
Agreed, slow play is an issue all the way from the highest pro level all the way down to the local league / weekly tourney at the poolroom. As a tournament director in a weekly tourney, there is no easy solution other than to stress to the players this is not what they do for a living, and to please keep it moving, for the benefit and enjoyment of everyone. If you have particular players you're having an issue with, it's always best approach them privately 1-on-1, between matches, to have a confidential conversation with them about the need for them to pick up their pace, as opposed to calling them out in front of everyone else as to their slow play. You might suggest that if they can't make a decision as to how to play a shot in 30 seconds, that they likely need to go ahead and play a safety.
As a TD in our weekly 9-ball tournament, I let all players know that in an extremely slow paced match, I have the discretion to shorten the race in the match, but generally I only like to enforce that policy in a match when the match is tied, and it must be before the hill-hil game. For instance, tied at 1-1 in a scheduled race to 4 I will shorten to a race to 3 before the third game starts. The players may not like it, but at least it's fair to both players to do it at that time.