This may be late-to-need; I see you posted this some time ago! I will caveat this post by saying that I am an avid fan of the PSX TSB (yes, I am the one). I still play on an old school PS2, and I have spreadsheets crammed with rosters and player attributes. I do not play TSBIII, but I am familiar with the player attributes because they are very similar to those found in the PSX TSB. In fact, I am currently playing a season using the All Star team roster from TSBIII.
Running Speed (RS) is the base speed of the player.
Rushing Power (RP) is something I am not entirely certain of, but I have some input. Popular opinion: time to accelerate/decelerate from RS to Maximum Speed (MS). My experience: this is the "blocking ability" of the player. The higher this is, the less likely it is that a player will get blocked. Case is point: Bob Nelson from the SNES TSB. Why was he so unstoppable in "the lurch?" Rushing power. It was the only attribute he had that could explain why he was an unstoppable force of nature. I have also noticed that the best pass rushers in PSX TSB tend to have above-average RP. Conversely, you want your run-stoppers and OLs to have LOW RP to ensure they don't miss blocks.
** You will want to flip RP and RS when converting players' attributes from SNES TSB to TSBIII. This should be fairly obvious when you compare the players attributes from both games.
Body Balance (BB) is the ability of a ballcarrier to execute a spin move when a tackle is attempted face-to-face.
Agility (AG) is the ability of a ballcarrier to execute a jump when a diving tackle is attempted from behind.
** Coming up with BB and AG are going to be tough sledding, as these were not attributes in the SNES TSB, so you will need to conjure these up. Finesse players tend to have higher AG (they run away from head-on contact), while Earl Campbell types have higher BB (they seek out contact). You will want your QB to have decent BB for that DL that pops though your line unexpectedly. Good KRs tend to have high BB (kickoff blocking tends to be sketchy in the PSX TSB ), and good PRs tend to have high AG (punt blocking is decent in the PSX TSB).
Passing Accuracy (PA) attributes in the SNES TSB are like ZIP codes in Seinfeld: meaningless. It was unused data. In later versions this was corrected. This represents the probability (multiplied by 100) of a QB completing a pass to a receiver who is in the vicinity of the pass. Case in point: in PSX TSB, if you intercept a ball thrown by Troy Aikman, better have it bronzed because he doesn't throw them. Ever. His PA = 81, and is the highest of all QBs.
Avoid Rush (AR) is the ability of a QB to complete a pass once a sack is imminent...I think. QBs with above-average AR seem to get an extra "beat" to deliver a pass when I am about to sack them--sometimes even while I am in the PROCESS of sacking them! Dan Marino (AR = 63) comes to mind. I can have that plodding Howitzer dead-to-rights, pass rusher diving to crush him, yet he manages to fire out a pass as if he has a force-field around him; my diving pass rusher falls harmlessly to the ground.
Coolness (CL) is how the QB will throw under a heavy pass russ. If it's low, he tends to get rattled and throws inaccurate/imprecise passes. If its high, he's Joe Cool (CL = 88 in TSBIII).
Kicking Accuracy (KA) controls the speed of the targeting needle’s oscillation. The higher the KA, the slower the needle will oscillate, making it easier to make an accurate kick. For punters, it is a ZIP code, if you will.
Avoid block (AB) is the same as AR for QBs, actuated during the act of punting vice passing.
Quickness (QKN) determines how well a defender covers a receiver. Low QKN guys blow assignments, are slow to react to routes, and are slow to run to passes underway. I'm not sure this was much of a factor in SNES TSB, but it has a profound effect in PSX TSB.
Keep in mind that the vast majority of my current knowledge stems from experience from the PSX TSB. Your results may vary!