They make great clothes hangers. /not helpful

I use the recumbent at the gym because I have a bad back too. Different brands have different seats, some of which are more comfortable than others. They have a pause button if you need to get up to pee, but Equinox disabled them in their branches. :-(

I'd search for a spinning bike. They have a flywheel and provide a kind of resistance that feels just good.
Because of the fact that for a lot of people they turn into clothes hangers you should be able to get one second hand for a very good price.
They are much better imo than the fitness pedal machines that gyms tend to have. They are more similar to real biking as well. Providing clips for your shoes f.i. Wear biking shorts to prevent chafing. So with that kind of machine you'll be well trained to go biking outside when the weather gets better.
Wrt bad back. It depends on what's wrong with your back. Ask your fysiotherapist. For some back ailments the problem is that the back muscles are too weak and the answer is to train your back muscles and then a recumbent bike might be not the right answer.
Wrt size; I haven't encountered a spinning bike that was too small for me. And some of them were Schwinns. Which is a US brand I think. So you and your boy should be allright since I'm 6'6".

Referring to your SO as 'your boy' feels a bit strange. Though it's your phrase. Thankfully english is a second language for me so I can feign tone deafness on the finer points of the socio-semantics of some expressions. He's not black I hope? :-)

The one we got (~$300) was difficult to adjust from one user to the next, and the power cable connector broke (so it would disconnect mid use) and there was no easy way to get a replacement cable.

Otherwise, it was fine!

Here's a string of ifs and ands, and a then: if you each have your own bike and if you don't mind a bit of assembly/adjustment every time you ride and if (big if, as I can't remember whether it's been sold/given away already) it's still sitting boxed and unused in my parents' basement after ten years and if you're willing to pay to ship it or pick it up -- it folds and fits into a rectangular box and weighs around 30 lbs. -- then you can have my Blackburn Tech Fluid Trainer.

It's probably not ideal, though, which is why it's still unused in a box in my dad's basement unless he or I gave it away to make room for other clutter. Sheer folly on my part to own it at all, I just like things with moving parts, but if you want it, it's yours!

I sat in a number of recumbent exercise bikes (because I have a rotten bad back, can't even sit on a non-recumbent at all, and needed a seat as comfortable as possible) and liked the Schwinn that's priced around $250 the best. I've had it 3/4 year now and it's working well for me. It had a brief issue (something went wrong with the cord) and they took care of it under warranty without question. Got it from Amazon w/free shipping. It's got an easy to adjust seat too.

You are not 5'1 no way.

Thanks for the info, guys!

And thanks for the offer, Hugh, lemme check it out.

jouke, I sometimes refer to my husband as "the boy" (not "my boy") in an affectionate way; it's just a speech affectation.

rhapsodie, I most certainly am 5'-1" (maybe even 5'-1 1/2"!). Did you think I was taller? Or... shorter?

I understood specklet. But thank you for explaining. :-)

I had one for a while that had a sort of fan thing as one of the wheels, like, it had fan blades inside a round cage so, I guess, you could make your own air conditioning. This was 10 years ago and it was probably 10 years old when I acquired it so this warning is probably pointless but still, do not get one like that! I suppose it would be excellent in a really hot place but the constant stream of freezing air on my legs was horrible.