That's strange -- the 9-volt Mk1 preamp is supposed to be plenty loud, especially compared to a passive Jazz.
Again, the clip I posted was recorded with a stock, unmodified Korean SR506 (so, with the Bartolini Mk1 pickups and preamp you seem to dislike).
But yes, there are different series of pickups in those basses. Taking into account only the recent-ish models (last 5 years or so):
- Lower-end SRs (SR400 and below) don't use Bartolini anything at all. They used to have Ibanez-made pickups and electronics, and were switched this year to "CAP" pickups, which I know nothing about.
- Middle-range SRs (SR500 to SR750, basically all the nice Korean/Indonesian ones) come with Bartolini Mk1 pickups (P-style split pickups in soapbar casings). They used to have Bartolini electronics as well, but they were switched to an Ibanez preamp.
- BTBs have Bartolini Mk2 pickups (soapbar humbuckers) and electronics.
- Prestige SRs (SR4000 and above, MIJ) have custom-designed Bartolini pickups based on the Mk2 and Ibanez 3-volt electronics. Having one such bass myself, I can assure you those pickups rock.
- Premium SRs (new this year) have Nordstrand pickups (Big Singles, which I suppose are ridiculously nice) and Ibanez electronics.
And of course, let's not discount the SRX line, whose higher-end models have meh electronics (2-band EQ) but nice-sounding and ridiculously hot pickups (Ibanez's own PFRs).
I did some digging for info on my old bass. I found some old pics. It was a SR500 supposed to be a prestige. That make any sense? It was definatley japanese made. It was a mahogany body with a bolt neck. 5 pc neck also mahogany. The stringers look like purpleheart but I cant recall. Oval abalone looking inlays. One piece bridge. Says SDGR on the headstock. Pickups do say MK1 right on them. 3 band pre but I dont know if it was Bart or Ibanez.

), SRs (MIJ) just don't sound well, that's it. Smb call them special, smb even say that this is an Ibanez character, but, obviously, if I can not put it in the simple mix in the next 30 min... into the oven!