I've sadly owned both the basses you have there. When I started learning bass 20 years back, I picked up the Jedson, because I thought it looked cool (vintage), however having played many basses since then I can say that was the worst bass I've ever owned - the pickups from memory were super low output.
The P bass, although being super lightweight ( a plus) is very low quality across the board.
If you want a decent short scale bass (especially as a beginner) I can't recommend a Squier Bronco bass highly enough, they are well made, the necks on them (I've owned half a dozen over the years) have all been excellent and super easy to play. You can pick them up around £80-100 and are far superior than the Jedson. The Bronco comes with a guitar pickup instead of a bass pickup, but don't let that put you off, it still sounds pretty decent.
I also know from experience that flatwound strings on the Squier Bronco sound great, yes, flatwound strings are expensive, but you will never have to replace them (they last 10 years plus) and are very smooth so are easier on your fingers, which makes learning and playing for longer much more fun. Some may shoot me down here, but I put full scale Fender flatwound strings (£25) on a Bronco once, and it made the bass come alive. So I would recommend them, and you will have a very decent playing and sounding beginner short scale bass.