Here are the specifics for special abilities to the best of my knowledge. I know this as part of common knowledge among fans of the Japanese Pawapuro series, which MLBPP is based on. I can't necessarily cite sources unless you read Japanese, so take it for what it's worth.

I could also be wrong on a couple of things if Konami or 2KSports decided to change things for this specific title.

I left out all the 2 abilities because they're the opposite of their 4 counter parts.

Positive Hitting/Fielding Abilities
Clutch Hit 4 - Contact and Power boost in RISP situations
VS Lefty 4 - Contact and Power boost versus lefties
Stealing 4 - Runner gets a better jump when taking off, returns more quickly to base during pickoff attempts.
Base Running 4 - Turns bases more efficiently
Throwing 4 - Throws less likely to be off-target
Consistency 4 - Affected less by negative condition (a player's condition is indicated by the face icons and affects hitting cursor size)
Durability - High injury resistance
Contact Hitter - Get line drives on off-center contact with the normal swing
Power Hitter - Easier to elevate pitches for fly balls with the big swing
Spray Hitter - Balls hit to opposite field more likely to stay in fair ground
Push Hitter - Loses less power when hitting to opposite field
Tough Out - Retain control over hitting cursor till the ball is closer to the plate
Gd Bunt - Less likely to pop-up bunts
Bunt Master - Even less likely to pop-up bunts
Firstball Hitter - Contact and power boost before the first strike of the at-bat (i.e. effect is still in place on a 2-0 count).
Gd Pinch Hitter - Contact and power boost as a pinch hitter
Table Setter - Contact and Power boost with none on
Tough Runner - Rams into catcher when crossing home plate. Catchers without Gd Block will be knocked over and lose the ball
Gd Block - Protects against runners with Tough Runner ability
Gd Catcher - Pitchers use less stamina and get extra control on their pitches
Grt Catcher - Even better version of Gd Catcher
Trash Talk - Diminishes opponent hitters' power.
Intimidator - Pitchers use more stamina when facing a hitter with this ability, and they lose some control when they face the hitter before them (thus the hitter before the one with Intimidator ability will see better pitches to hit).
Hot Hitter - Contact boost after getting 1st hit of the game.
Walk-off Hitter - Contact and/or Power boost (There are actually THREE abilities by the same name occupying same square. One is for walk off hits and the second is for walk off homers, the third is a combination of both. You can confirm this in player edit mode) in walk-off situations
BL Hitter - Contact and/or Power boost (There are 3 abilities by the same name, for getting hits, homers and both, similar to Walk-off Hitter) in walk-off situations
Good B2B Hitter - Contact boost when the previous hitter gets a hit. (Must be in the same inning)
Rally Hitter - Contact and Power boost when team is behind
Refined - Even when the hitter makes poor/weak contact with a ball, it will have a better chance of staying in fair ground and also fall in for bloop hits more often.
Gd Morale - Entire team gets ability boost when player is on the bench or playing
Spider Catch - Can climb outfield walls

Neutral Hitting/Fielding Abilities
Sparkplug - Depending on player's performance for a particular game, the rest of the team will get an ability boost (similar to Gd Morale) or reduction (Pr Morale)

Negative Hitting/Fielding Abilities
Free Swinger - Contact decreases after two strikes. (Returns to normal on a full count)
Error Prone - Player's E FLD reduced in opponent's RISP situations

Positive Pitching Abilities
Poise 4 - Less likely to be rattled (see stars) after getting knocked around. Less affected by hitters with Intimidator ability
w/RISP 4 - Velocity and movement improves with runners in scoring position
VS LH Batter 4 - Velocity and movement improves versus lefty batters
Fastball Life 4 - Fastball reaches the plate faster than the radar gun readings indicate
Spin 4 - Sharper break on all pitches with movement
Release 4 - Opposing runners lose 1 from their Stealing ability (ie Stealing 3 becomes a Stealing 2)
Consistency 4 - Affected less by negative condition (a pitcher's condition is indicated by the face icons and affects top speed and amount of break on pitches)
Recovery 4 - Quicker recovery during Season Mode
Gd Reflexes - Better chance of fielding comebackers
Gd Pickoff - Faster pickoff throws
Battler - Does a fist pump when certain conditions are met in a game, after which stamina usage decreases for a while. I've also heard having this ability cancels out hitters' Intimidator ability.
Gd Low Pitch - Control bonus on low pitches
Gd Release - Same arm action on fastballs and breaking balls.
Groundball P - Diminishes hitters power
STR Finisher - Velocity and movement boost in late (7-9th) innings (only affects pitchers who start the specific game)
Gyroball - 4SFB will have a bullet spin and downward break, plus it will reach the plate faster than a non-Gyroball with the same radar gun reading (similar to Fastball Life 4 but even more powerful)
Lucky - Team's hitters will get an ability boost when the pitcher is in the game (= good run support)
Intimidator - Diminishes hitter's power, and in the case of pitchers coming on in relief (applies to pitchers classified as "Starter" in their stat screen IF they are used as relievers for that particular game), contact as well. The shrinking hitting cursor should be fairly easy to notice when a closer with this ability takes the mound.
Safe Pitch - Mistake pitches will not be down the middle
Dr. K - Increased fastball life and spin after 2 strikes on the hitter. Because of how Free Swinger loses its effect on a full count (you can confirm the cursor size returning to normal), I'm hypothesizing that Dr K doesn't work on a full count either.

Neutral Pitching Abilities
Poker Face - Will not pant after running out of stamina, or get dizzy (see stars) after getting knocked around. The actual effects like diminished velocity/movement will still be there. The only difference is the pitcher will not show it on his face.

Negative Pitching Abilities
Choke Artist - Velocity and movement decreases when 1) Pitching to close out a win either as a starter or reliever; or 2) Pitching in the 5th as a starter with a lead (after which the pitcher will be in line for the win, hence a "choke" situation)
Walk - Control decreases after 3 balls. Because of how Free Swinger loses its effect on a full count (you can confirm the cursor size returning to normal), I'm hypothesizing that Walk doesn't work on a full count either.
Hothead - Will not get the dizzying effect (see stars) after being knocked around, and instead blow steam off the top of their head. Unlike pitchers seeing stars, pitchers blowing steam will not lose velocity or movement on their pitches, nor get drained of stamina quickly. They just lose command. For this reason it's often considered a positive ability despite being in red.
Unlucky -Team's hitters will get an ability reduction when the pitcher is in the game (=poor run support). After an Unlucky pitcher leaves the game with a lead in line for the win, the opposing team's hitters will get an ability boost. (=blown leads by the pen)
Slow Starter - Velocity and movement start out decreased before gradually returning to normal over the first three innings.
Shuuto Spin - 4SFB will have occasional arm-side run and hitters get a power boost when that happens
Sandbag - Velocity and movement decreases when pitching to the bottom of the lineup.

no spider catch is climbing the way

go into practice mode and press 2 on the wii, and whoever has the ability he will climb the wall and then do a little jump

(idk what button on ps2)

Not true. It's in the manual and any player with or without the ability can do it with the jump.