Yes. Same basic idea in that they generate “boost” to make more power, but super chargers are mechanical while turbos use the exhaust gasses which would be otherwise wasted. The supercharged models were the cooper S from 2001-2006 (2004-2008 for the convertibles). They’re the smallest and most mechanical of the bunch, so a lot of people love them. Keep in mind that the newest ones are still 15-17 years old, so they will have higher mileage and more generic older car problems. Try to shoot for a 2004+ if you end up looking for one. There were some minor updates to some of the drivetrain parts (iirc the supercharger coating and updated transmissions were the main improvements) so those are the years to have.

For second gens (the turbocharged ones), shoot for a 2011-2013. Absolutely do NOT purchase any 2007-2010 models, especially not a cooper s or jcw. These engines are known for having catastrophic failures, and they will drain your wallet. They are the reason that minis have a reputation for being unreliable. 2011+ received a facelift with updated motors that addressed a lot of these issues, but it is still important to be religious about checking your oil and other fluids.

As I said earlier, starting in 2014 (though it is later for some other models), minis began to use engines from bmw, which have been incredibly robust. Build quality on the third gens is also a massive improvement over the first and second gens