I haven't lived in the States in a long time, and am learning as I go.
OK. So with the standard disclaimer (I'm not an eldercare expert, I just play one on the internet ;) ), and apologies for this being pretty basic (and yet also slightly oversimplified, i.e. I'm focusing on the big points), here's an overview:
Medicare is a health-insurance program that most Americans 65 and older qualify for; the costs of coverage are minimal to the insured (e.g. your mom) compared with other policies offering similar benefits; it is standardized across the US.
Medicaid is a means-tested program that, among others, a disabled older adult (someone in need of long-term care, for instance) would qualify for if they met the financial requirements (i.e. were poor). The details of what Medicaid covers vary state-by-state (it is a state program that is subsidized by the federal government and must meet certain federal requirements -- minimums -- but not every state covers the same stuff beyond the minimums). By the way the five-year lookback period you mention pertains to money given away to (or spent on) others, but not to expenditures on herself. So if she blows every penny to take a world tour, has a horrid accident, and requires long-term nursing home care, she is OK (in terms of Medicaid coverage), but if she blows every penny to take herself, you, and the grandkids on that world tour, then you all would have to pay back your share (to Medicaid, basically) before she would qualify. The point, of course, is to prevent people from qualifying for Medicaid by handing their resources over to family members.
Medicare has multiple parts. Part A covers inpatient care (in a hospital) and there is no premium required to receive it. Part B covers 80% of outpatient care (the patient pays the rest) and the premium is about $100/month. Part C isn't what most people use (an alternative to B) and I won't cover it here. Part D provides prescription drug coverage and also involves a monthly premium and copayments, which can vary.
Feeling glad you moved to France? Where were we?
Ah yes, Medigap plans, also federally regulated, are private insurance that covers most of the 20% not covered by Part B. Costs vary; I know my dad's policy runs him about $200/month (although he is on Medicaid, he is allowed to pay for the Medigap policy from his social security because, obviously, on average this saves the state money in the long run).
Medicare covers short-term (up to 3 months, I believe) skilled nursing care (nursing home residence) if a covered person has a medical problem (heart attack, broken hip...) that requires at least a 3-day hospital stay and a need for rehab (physical therapy) so the patient can regain independence. Other than that, it doesn't cover nursing home care, only Medicaid does.
A month in a nursing home right now costs about $6K. Long-term care policies help meet those costs; my impression is that some such policies are much more "complete" than others. Complete doesn't necessarily mean better -- like any insurance, the more you demand, the more you pay. Taking on more of the risk yourself results in a lower premium and may prove a better value.
So, that's an overview and some numbers. Where I'd recommend you start, in terms of yourself, is thinking about what you would and wouldn't be willing to do for your mom. As my original post in this thread hinted, for me I think this is partly a matter of circumstances (medical crisis versus gradual decline, expected recovery versus not) and it helps me to think about particular and plausible circumstances, but really, on some level the issue here is what do you want to insure
yourself against. What kinds of assistance would you feel obligated/motivated to provide your mom yet be unable to do so? Those are the ones you want to plan for (by saving, by buying insurance, whatever...). I'd also recommend both that you educate yourself (more) on the US system. Indeed, I'd guess you might quite likely be well advised to pay for an hour or two of the time of someone familiar with US insurance products for people like your mom and to get their advice on how best to allocate the resources you have available to commit for this purpose (obviously someone who doesn't benefit from selling you something will be the better choice here, and my assumption is that there are any number of certified financial planners who charge hourly fees and don't sell insurance but still advise their clients on what sorts of products to buy).
In terms of talking with your mom, honestly I wouldn't start with anything approximating money or budget. I'd talk about what she wants her life to be like in the next few decades (realistically "the rest of her life," but you may not want to phrase it that way) and use that as a way to ease in some suggestions about things she may want to consider to make that possible. Certainly there are ways to bring money, and savings, and so forth into this, but there are lots of other planning issues that relate to the financial without being directly financial. Has she thought ahead to things like how she would manage if she were unable to drive either for a half a year or for the rest of her life? How could she plan for that? Is she connected to social networks within her community that she could turn to for help (church? canasta club?)? If she had a medical crisis, who would have access to her information and know what bills to pay and from where?
As an ice-breaker (for the above conversation), my understanding is that one woman in five will have a broken hip in her lifetime -- pretty amazingly high number, IMO. So this doesn't need to be all about horrid diseases and death and stuff, it can be "just" about things that are (plausibly) likely and unpleasant, but usually surviveable and manageable with some foresight.
Good luck; this is tough stuff. I'm sorry you're dealing with it and honestly, believe that you will be doing your mother a favor if you are willing to talk through this stuff with her. Regardless of how much financial help you do or don't provide, it's good to know what she wants if you may be in a position of needing to make decisions on her behalf.