The problem is that several things are getting conflated here. There's the way certain terms are used medically, and the way they're used colloquially.
Medically speaking, a lot of viruses produce the same symptoms, because your body fights them all off the same way, at least initially. You get congested, may have a cough, may have a fever. We usually refer to those symptoms as "having a cold."
There is a strain of viruses that are known as influenza viruses; depending on the person, the strain of virus, and much else, when you get infected with an influenza virus, you may experience symptoms like you do when you "have a cold", or you may experience worse symptoms. In general, if you progress to things like body aches, vomiting, and so on, you'd probably refer to that as "having the flu."
And now there's COVID-19. For most people, if you get infected with COVID-19, you have no symptoms. Some people will have cold-like symptoms, some people will have flu-like symptoms, and some people will have symptoms that are, for the most part, unique to COVID-19.
Until very, very recently, we usually didn't test for any of the specific viruses that caused cold or the flu, because it didn't matter. We had no way of treating viral illnesses; the focus was on addressing the symptoms and making the patient feel better (we'll ignore over-prescription of antibiotics here). Doctors might test for strep throat, because that could be treated with antibiotics, but in general, if you came to a doctor and had cold-like symptoms or flu-like symptoms, you'd be given a fairly generic set of recommendations. We didn't really have a good way to test for viral illnesses anyway; things like PCR testing was incredibly expensive, and again, it didn't matter because knowing what virus was causing your illness wouldn't change how you were being treated.
Even if you were hospitalized, it didn't matter what you had. That's changed, recently, as we have some medications where it really matters if you have a cold virus, or an influenza virus, or COVID-19. And PCR testing is much, much less expensive than it used to be.
At least where I am, if you were sick enough that you'd stay home from work (and this was a problem, but no matter), you'd say you had/have the flu. You would go work with a cold. What you'd call your illness was really a question of symptoms, not of what was actually causing the symptoms.
To answer the question, unless someone had specifically told you "I have a cold," or "I had the flu," or whatever, I'd just ask them "Are you feeling better?"