as a recording engineer that has recorded and mixed ... over a thousand songs, i can share my experience:
1) yes, the player is the main thing. not so much whether they are amazing players. but how much experience they have with recording. things like finger/string noise are a whole different issue in the studio vs on stage. it takes a lot more time to edit out string noise and in the studio time is money. the less labor intensive approach is to throw more processing at it -- like a de-esser, multi-band dynamics processor, dynamic eq or transient designer (behind a crossover) to try and deal with it. this kind of processing is hard to do without radically changing the character/dynamics of the performance.
2) often times, cutting a bass track that sounds good solo'd isn't the issue, it's setting up the mix engineer to be able to get the important qualities of the bass part to sit in the mix. the most cluttered frequency range in any mix is the midrange and in particular getting the right balance in the lower midrange (250Hz - 700Hz) is really key. for bass, you need a solid low end that somehow dances with the kick (and any low synth stuff) along with the part that translates on tiny speakers which is typically just above the lower mids (600 - 850 or so). for this reason, i think basses with multiple pickups are a bit trickier to capture in such a way that they will retain what the mix engineer needs. depending on how the two pickups may be balanced there's some phase cancelation (you can look up the 3:1 rule that comes into play when recording drums) and as a result the frequency spectrum is kinda choppy vs. a single pickup which is like using a single microphone on drum kit -- everything is represented because nothing is being cancelled out by combining multiple mics that have slightly different phase variations on the kick or snare. with J basses it's kind of part of the sound we are used too -- a bit hollow -- can be a bit aggressive -- but retains the lows and the highs. again, that midrange can suffer in such a way that all the processing in the world won't give you the stuff you need in 600 - 850 to get the melodic aspect to sit in a mix.
3) the issue with active instruments can be headroom. not so much needing or wanting more or less but how the preamp and the resulting chain be it amp or DI reacts to the output. every buffer has a "tone" that interacts with everything else. some chains will react with a "splat" that might come across live at all. so in the studio, it's always nice to have the option to bypass the active or use a passive bass in case you run into this.
i have a whole bunch of basses in the studio including a bunch of P's because i also like to be able to offer things that help us get closer to what we're going for with each tune with less processing. i want the rough mix to have the character we want and not leave it to the mix for it to all gel.
so my simple answer:
for finger tone, i think it's hard to go wrong with either a stingray or precision assuming you get a nice clean recording without clipping (in the signal chain) or excessive hum.
for pick, you'll want to be careful about those transients with the ray. and you can't go wrong with a P.
for slap, i think the ray would prob win but again, listen to the transients and look at the waveform -- look for any squaring of the transients -- meaning, it's not enough to make sure you don't clip the converters on the way in -- the red lights won't tell you if you're clipping within the signal chain (i.e., a pedal or amp or DI).
one more thing: do your mix engineer a favor and always mult and record a completely dry DI signal (that doesn't clip). they can use this to reconstruct any string-to-string inconsistency and also use it to "key" other dynamics processors. i always set up dynamics to duck the bass when the slap of the Kick hits or Snare or a Vocal transient hits. that way i can keep the bass LOUD and it doesn't fight with the K, S or Vox.
sorry to go on and on but this is the stuff i live and breath and i'm always happy to help anyone with any studio challenges so feel free to PM.
-tE