Many of the (continental?) European "every-day" bikes have a relatively upright and straight back position (you basically get a continuum there from sportive strongly forward tilted position to upright or even slightly back tilted.)
Typical features of such a bike with more upright/straight back position are U (or M) shaped handle bars (the ends are somewhere between 45° to the outside and parallel to the bike; they may slope downwards), the handle bar is often above the saddle and the stem doesn't go much forward*:
While many of these have rather heavy frames (particularly the "grandma frames" with single tube with very low step through need that one tube to be very thick), there are also lighter ones. E.g. as a student I rode one with a light aluminum frame: it was a diamond frame with two thin tubes for the top.
The ones with very upright sitting position often have extra springy saddles and there's also the possibility to have a suspension seatpost. And of course wide tires run at relatively low pressure. On the front, the fork of these bikes will usually provide some suspension as well.
An acquainance with back trouble uses a trekking/trouring bike and a butterfly handlebar that is turned upwards (and the brakes are on the upper part). Somewhat similarly (but not for long term use), I once needed a more upright position on a tour and the solution was turning my bullhorns upwards (and change left-right, otherwise they'd point inward back).
I've also seen roadbike handlebars mounted upside down, but I cannot say how well that works.
Others have already written on trikes and recumbent bikes.
Jahaziel correctly points out that among the bikes with this easy or comfort geometry there are many "tanks" that easily weigh 20 kg. However, in my experience, there is substantial variation in that, and I suspect that some of the options that help making modern bikes lighter may not be good for OP's back, e.g. the straight aluminum forks that have no suspension whatsoever. And an aluminum frame with suspension fork can easily weigh as much as a decent steel frame.
OTOH, I've seen (and owned) bikes that were more sportive than the indestructible 3 gear tanks, but with a more relaxed/upright position than on a road bike, with somewhat lighter frame and derailleur gear.
With a lifting capacity of 5 kg, OP will need help to get up the bike when it is lying on the ground for all but the most naked road bikes.
I further suspect that there are further "weighty" decisions that only OP can take:
- is the softer riding on balloon tires worth additional 2 - 3 kg?
- is the weight of rack + basket a good investment since OP cannot carry a backpack, and bike bags are heavier than a thin normal bag in the basket?
- Battery lights instead of a hub dynamo?
- derailleur gears instead of internal gear hub? What range of gears does OP need?
- Or is sturdiness important after all since OP needs to be able to rely on the bike?
- Will a light bike frame + appropriate lock still weigh less than an old, not too heavy steel frame bike which can be left around with a light cable lock?
- Butterfly and those U-shaped handlebars do weigh more than a narrow straight one of course (BTW, they are available in aluminum, no need to have them in steel as in the image). But that doesn't help OP if the required position is not possible with the light handle bar.
Putting a different stem and handlebar on a light trekking bike frame may go a long way (but that will need careful trying - but then, OP will probably need to carefully try a whole lot of bikes/part combinations).