Actually, whether or not you get a shield bonus is dependent on how many shields the square would provide and whether or not you have an industrious civilization.
If the square would provide no shields, then you get 1 shield in the center square regardless of civ. At size 13, an industrious civ would get two shields. If the square would provide 1 shield, then you get 1 shield. If you have an industrious civ, at size 7 you get 2 shields, at size 13 you get 3 shields. If the square would provide two or more shields, you would get those and add one more for industrious civs at size 7 and 13.
If the square is near fresh water, then an Agricultural civ will get 3 food in the center square. This is true regardless of lake or river.
If you have a Seafaring Civ, then you get +1 commerce in the center square if and only if you are on the coast. If you have a commercial civ, you get +1 commerce in the center square regardless of where it is. Not sure what happens if you are both (i.e. England).
All shield and commerce bonuses apply to the center square. No food bonuses apply to the center square. Note that you get one 'free' shield in the center square, however, if you would already get a shield the 'free' one does not get added. If you build on a hill with iron, you will get two shields in the center square. In addition, if you build on plains/forest with furs you will get two shields in the center square. There are other ways to get shields there as well (uranium and coal, for instance) but this should give you a good idea how everything works.
The center square will always give at least 1 shield, 2 food, 1 commerce. This is true even if you mod the scenario to allow cities on mountains.
