Steering away from Fenders is of course a matter of taste - I did the same for years but I still can't offer any advice since your taste is unlikely to be identical to mine.
As to what 'lightweight' means, a typical Fender Precision comes in around 9.5 lbs. It's a bell curve, and you can find P-basses that weigh 11 lbs and P-basses that weigh less than 8 lbs, but IME that 9.5 figure is where the great majority of them sit.
In my 20s I would have thought of that as a lightweight bass. Now in my 60s, that's pretty much the top end of my range.
A typical Fender Jazz is substantially heavier than a Precision - that offset contoured body is way bigger.
My feeling is that anything under 8 lbs can fairly be described as 'lightweight', but don't forget that balance is crucial to the equation, and that your choice of strap is also very important.
You can often save weight by going hollow-bodied or semi-acoustic ... a genuine German-made Hofner will usually weigh around 6 lbs, a Fender Thinline or similar (e.g. the Grosmann Classic Semi-Acoustic currently in the For Sale section) will be around 7.5 lbs.
You can also save by going futuristic and minimalist ... check out Steinberger and similar.
Above all, never lose sight of the fact that bad posture does WAY more damage than the weight of your bass ever could do.