I've been a captain in the APA for a while and here's how I define things:
2 (and as a point of information, in manhattan, where I play, as with the national format, only women can be 2's): This is going to be someone that can pocket a ball or two, but really has no hope of playing both a good shot and good position. It's a war of attrition/luck for these players to win a game.
I feel the transition from a 2 to a 3 is either when the player gets good enough at pocketing to be able to make more than just one shot in a row reliably, or if some safety play is learned and incorporated - either of these two advancements will raise a player from a 2 to a 3.
3: Female 3's can generally hold a cue decently and would be able to run a couple of balls and understand a bit about where the cue is going to go even if they can't make it happen all the time. Men will range from absolute beginners (since they can no be 2's) to being able to make a few shots.
I think the transition from 3 to 4 is when your fundamentals start to come together a bit and you have played enough to understand how the cue can be moved around (draw/follow/spin).
4: These players usually are starting to grasp all the fundamentals together. They have a semi-decent stance, bridge, form. They understand how to use follow and draw and can execute them inconsistently. Given an easy open table they should be good to run 3-4 balls. They do not have the control to really run out or play fantastic safeties. They will not think very far ahead.
I think the transition here is when you can play at least a moderate sized area position consistently. 4's will go up to 5's when they can put the cue ball somewhat near where they want it to go as well as executing pocketable shots consistently.
5: In my league these players can be the wildcard. A weak 5 is not going to be very hard to beat. But a strong 5 might be able to manage a break and run here and there, will probably be able to play a tough safety to get out of. To me the biggest thing that identifies 5's is the amazing ability, time and time again for them to run until they have just 1 problem ball left. The full rack run out though process is not learned here yet. They can move the ball around, break things up, but not well enough to really handle all the situations that come up. More often then not a 5 will clear most of the table for me and leave an easy run out, even if they are shooting well.
The big thing here is the ability to complete the rack. Knowing what racks wont be likely to be completed without exceptional play. Having an ability to honestly calculate the percentage to make a shot/run out and either go for the out or play an appropriate safety at the appropriate time. Also at this point you should be building an arsenal of "difficult" shots to call on. IE they have played enough to know the details on how to make shots that are not obvious/require solid fundamentals and have played enough or practiced enough to actually make them consistently.
6: 6's should be able to run an open table (that is one with no major problems) and should be able to run out on tables with only moderate problems (say a cluster to break up, but a favorable break ball already in position). They should at this point be able to calculate more advanced shots - lining up off angle combinations. Knowing how to aim bank shots and make them at least somewhat consistently (well the "easy" ones anyway). They should have some understanding of how to move the cue around the table even if they have to use multiple rails. Safety play should be to the point where they can lock up the cue for any lesser handicapped player sufficiently.
The basic difference here is that 7's are just better at everything than the 6's. They will make every shot a higher percentage of the time. Be able to execute a break and run more often and on more challenging rack layouts. They will play better safeties. They will play better position. Etc etc etc. I don't think many 6's watch a 7 play and are unable to see what's going on, they can imagine it, just maybe not execute it as well.
7: These players run a wide gamut as already mentioned. In particular newly minted 7's usually do terribly against strong 5's and 6's. Since the way the game spot works when you move to a 7 everyone else has to win one less. This makes things much easier for strong 5's and 6's. On the flip side 7's can run all the way up to open speed players. The APA manual says you can't play in the APA only if you make a significant amount of money from pool - well that rules out just about everyone - including plenty of semi-pros. There are plenty of 7's I (as a strong 6 in my league) absolutely do not want to play because the odds of my winning are very small. These players at least should have a chance to run out most racks that aren't overly challenging. And should be able to both play very good safeties and get out of most moderately well played safeties. A 7 should not be giving up ball in hand in any but the most difficult circumstances (or when you give it up as a safety in and of itself). Personally I think this is a huge weak spot in the handicapping - there just aren't enough levels to represent the skill range in the 7 category...
All that being said - if you search for previous posts by me you'll see my big rant on the APA from another thread
I think it is pretty easy to put someone in the right category just by watching them play a few matches - you can sandbag all you want, but you simply can't hide fundamentals. A skilled observer can easily notice when someone is under handicapped.
and 1 time I made semi's. I'm pretty sure my Handicap - at least in respect to our local competition around here is very accurate.