Occasionally, I have been accused on providing a definition of semi-presidentialism but not of any other regime types. So, here are the definitions I work with:
Semi-presidentialism – where there is a directly elected (or popularly elected) fixed-term president and a prime minister and cabinet who are collectively responsible to the legislature;
Presidentialism – where there is a directly elected (or popularly elected) fixed-term president and where cabinet members are not collectively responsible to the legislature;
Parliamentarism – where there is either a monarch or an indirectly elected president and where the prime minister and cabinet are collectively responsible to the legislature.
Monarchy – where there is a monarch and where the cabinet is not responsible to the legislature
The phrase ‘popularly elected’ is meant to capture US-style and former Finnish-style electoral colleges.
My philosophy about how regimes should be defined can be found in Robert Elgie, ‘The classification of democratic regime types: Conceptual ambiguity and contestable assumptions’, European Journal of Political Research, 33: 219–238, 1998. Basically, I define regimes on the basis of dispositional rather than relational qualities. One way of understanding this distinction is to say that I classify countries on the basis of objective constitutional criteria (is there popular election, yes or no?), rather than subjective political criteria (is the president powerful?, or is the country a democracy?).