Locus standi to answer: Been practising yoga for 11 years, on and off, in different programs. I'm Indian so a lot of texts and treatises, including some of the original ones are somewhat accessible to me.
Now first off, the idea that there are different 'forms' of yoga is almost certainly hogwash. There are different schools, sure, and that's only because different teachers specialise in different areas and try to develop those areas more. I'll come to this in a little more detail later.
Second, you needn't worry (or be dismissive of) 'eastern philosophies', because you will anyway NOT be able to integrate those into your yoga practice until years. Any teacher who starts talking about energy flow and stuff to a beginner class is very likely taking people for a ride because they don't have means to verify.
Now, coming to the dangers involved - they are very real if yoga is practised incorrectly or under a substandard teacher. I would strongly recommend against learning online. A good yoga instructor is trained in anatomy, to the extent that they will be able to tell when you are in pain, when you are using the wrong muscles or breathing wrong by looking at your posture. You won't get that online or with a teacher who has done a 2-month training course.
Coming now to a suggestion : try Iyengar yoga. Its a school of yoga founded by BKS Iyengar, and based on information provided in your question, I think it might be best suited for you. Reasons:
(1) BKS Iyengar was an extremely ill, un-flexible child who was forced into yoga. So his whole idea is to get people who are less flexible/aged/injured into yoga by means of props where necessary. That's why you see a lot of bricks, bands and ropes in Iyengar yoga classes. Your physical limitations are actively taken into account.
(2) Instructors need to train rigorously for a long period ( I think 7 years or so) before they can start teaching. This included periodic stints at Iyengar's house for personal training by Iyengar (while he was alive, RIP). So it is highly regimented, most wishy-washy students will never make it to instructor.
(3) It looks at holistic fitness and flexibility, not extreme spine bending, weight loss and stuff. If you learn long enough, you may get to some jaw dropping level, but for the most part, its gentle and gradual, but still rigorous.
Dislcaimer: I am not partisan to this school, it just seems best given the OP's question. One school I would recommend against for a beginner is what is called 'Ashtanga/Vinyasa', that is usually a little advanced and requires a fair amount of flexibility to start with.