I call them mountain lions, normally just "lions". I have called them lions my entire life. A lot of people call them cougars, and even though the ones in Florida are referred to as panthers I still call them lions.
The USFWS has brought in Texas strain Mountain Lion to increase genetic diversity in the Florida strain mountain lion population. So even if the ones in Florida were a isolated patch of cats, they are now related to the ones in South Texas.
Here in New Mexico some people call them cougars, and some call them lions. It is very regional (down to the county) depending on where you live in the Western USA, and Canada. Most Canadians also can't figure out what to call them either.
State game and fish agencies refer to them officially by a variety of names.
Arizona Mountain Lion
Texas Mountain Lion
Wyoming Mountain Lion
Montana Mountain Lion
Nevada Mountain Lions
Idaho Mountain Lion
South Dakota Mountain Lion
North Dakota Mountain Lion
Colorado Mountain Lion
Oklahoma Mountain Lion
Nebraska Mountain Lion
Utah Cougar
Alberta Cougar
New Mexico Cougar
Yukon Territory Cougars
British Columbia Cougar
Oregon Cougar
Alaska both cougar and mountain lion
California both cougar and mountain lion
Florida Panther
I am from Wyoming originally, and will always call them mountain lions.