I'm kind of in between.
I agree that with no other context, just in a context like journalism, 'rehab' in English is likely to be assumed to mean drug or alcohol rehab.
I also agree that certain other typical kinds of medical rehab usually have another word added, like stroke rehab or cardiac rehab.
On the other hand, a growing segment of the US health-care industry is actually rehab hospitals, which cater specifically to people recovering from things like major surgery, car accidents, traumatic brain injuries, and so on. (Because treatment for those acute conditions is typically covered by insurance, whether public or private.)
And rehab is a very specific, defined term in the context of skilled-nursing facilities (aka nursing homes), where (as we learned to our dismay with a dementia patient) physical, occupational, and speech therapy are typically covered by Medicare only for a certain number of weeks and only if the patient is able to complete a certain number of hours per day or week of fairly intensive rehab sessions with a therapist and demonstrate continued 'progress' by meeting certain criteria. (More recently there has been a court case, Jimmo v. Sebelius (2013), that was supposed to ensure that ongoing, long-term wellness therapy is covered precisely for chronic conditions like Parkinson's and dementia. But many facilities still apparently aren't aware of it, and it may not be consistently enforced or even acknowledged.) If an elderly patient is not able to meet the criteria for 'rehab,' he or she is considered a 'long-term care' patient, which means that nursing-home fees from that point on, typically $9,000-$10,000 a month or more, are not covered at all by Medicare after a certain length of time such as 100 days, unless he or she is first re-hospitalized for at least a three-night stay and then approved for rehab for that new acute condition. That obviously has huge financial implications for families.
So, I don't know how to line up those contexts with Reha or other German terms, but I wanted to at least get them into the discussion.