Gee, what fun!
Lessee, you can call it what you want, and in Thailand you're probably more right than I am, but in the USA the two ways most people get to the airport (if they don't go in a private car) are the Airport Limo (usually a large van), or a taxi. The difference is the limo makes several stops on the way to pick up people, and the cost is a little less than half of what a cab (taxi) would cost. I'm sure there are many other ways to get to the airport including helicopters and huge Cadillac limousines, but the most common are the first two.
I realize there are lots of ways to get from the Phuket Airport to Patong Beach, but for a visitor, the choices are usually one of the two readily available when you leave the terminal - what you're calling a mini-bus, and a taxi. The first is 150 baht, involves being squeezed in with lots of other people, and a stop part way to Patong where you might change vehicles, and are asked if you want to change hotels to one they can provide. The second is the taxi. I know (now) that there are at least two more choices, if not more, but for most people the choice seems to again be one of the first two, which you get to choose from as you're walking out of the airport terminal.
Nice to see all the piccys posted up above. The van (mini-bus) looks like a smaller version of what we've got in the US. Nowadays many people call it a Shuttle, and the company Super Shuttle are moving into many airports all over the country. It's a less expensive and very reliable way of getting to the airport. The "Taximeter" photo looks like a regular taxi. The Limo in the third photo looks like a regular street car - a Limousine in the USA is usually assumed to be quite long (sometimes extremely long), with a very fancy interior. They're usually pretty expensive to hire. Most of the time they used to be Cadillacs or Lincolns, but nowadays you can find many high-tech cars that have been converted into Limos - but these are usually one-of-a-kind vehicles.
The photo of the big white "Limo" posted above would be quite an eye-stopper in the USA. It's made from a Corvette, and I've never seen one of these before, ever. Since all the cars in the photo are Corvettes, I'm guessing this is a show car, designed for attention. It's probably the only one of its kind in existence!
Limousines.... go to
Google and select "images" at the top line. Enter the word "limousine" and do a search. What you see there is what I think of when I hear the word "limousine". I doubt if one of these ever has, or ever will, be available for transport from Phuket Airport to Patong Beach.
We might all be speaking English here, but it's obviously different depending on where we live(d), and so on. Not even sure why I'm posting this now, other than that it's a nasty storm outside and I couldn't sleep.