If you went to study French in France

Less than 90 days and not in a degree-granting program means you should be fine as a tourist. Just be careful to get out within those 90 days; you wouldn't want to have issues down the road!

They should understand price - what are you proposing to pay for - A 10 week course, or a Year long course? For less than 90 days they won't see an issue because they won't have one.

If it's not a program that grants a degree then you are fine on a tourist visa as long as you stay less than 90 days.
I have no idea about the school's medical requirement. Are they requiring that you purchase their insurance or is it just a suggestion? Most schools like this just require proof of insurance for civil damage or medical costs. Minor medical care in France is relatively cheap. People usually buy travel insurance to cover big expenses such as medical evacuation or hospitalization.
Usually when attending a degree-granting institution in Europe, they give you an ID number that you submit to the consulate in order to get your student visa. You shouldn't need this though since you aren't going to be enrolling in a degree program.

It’s bizarre that Alliance Française of all places can’t tell you about visa requirements.

I am not so sure as the others that you don’t require a visa, but you should look at the website of the French consulate in the U.S. to know for sure.

Our daughter studied in Perpignon for about 6 weeks one summer and was there with no student visa. So I think you should be fine.

As long as you stay for fewer than 90 days, and as long as you don't take a job, and as long as you obey the law, it wouldn't matter if you stood on your head for 8 to 10 weeks or spent your money at the AF in Lyon for 8 to 10 weeks. You would be on a visitor's visa.

By the way, if you do go to the AF in Lyon, be sure to stop by the nearby Bahadourian flagship store. It's an institution. There's one at Les Halles Paul Bocuse, but the one just off rue Paul Bert is special.

No one would care about your medical coverage because you wouldn't be seeking a titre de séjour; rather, you would be visiting as a tourist for under 90 days in the Schengen zone. In other words, you're on your own for medical coverage.

Another way to look at it...as long as you respect the 90 Day limit, if the School does not require proof of a Visa, then the Government will not care. For Degree type course, the School will want proof of Visa before starting, Immigration is oblivious to your needs, often you need to make an effort to get the proper stamp or visa validated.

Honestly, it's not that surprising that the AF didn't give much visa advice. They have students from around the world in different situations, some living locally long-term working or with a working spouse, others coming for short-term studies, and they are not a degree-granting institution. It's not their job to know what kind of visa students need in the same way that a university would get involved (which again, in this case, one is not needed). Plus, they probably don't want to give someone the wrong advice since it can be so different depending on national origin.

Thanks everyone.
I decided on Lyon for a month. Strasbourg for a month. Then a few weeks to travel around, leaving on the 87th day! That should be safe enough, I hope.

The problem I am facing with schools in both areas, is that they have lots of holidays, and even stop after quite early in April, I think the 17th, for a couple weeks break! Anyway, I will sort something out. But so far, I am pleased that in both regions, there are opportunities for day, or weekend trips.