Since in my opinion you need to go to the OED to understand the English language and its origins, here is the definition from that source:
saddle


• noun 1 a seat with a raised ridge at the front and back, fastened on the back of a horse for riding. 2 a seat on a bicycle or motorcycle. 3 a low part of a hill or mountain ridge between two higher points or peaks. 4 the lower part of the back in a mammal or fowl. 5 a joint of meat consisting of the two loins.
• verb 1 put a saddle on (a horse). 2 (be saddled with) be burdened with (a responsibility or task).
— PHRASES in the saddle 1 on horseback. 2 in a position of control or responsibility.
— ORIGIN Old English.


Note that the important criteria is the raised ridge front and back: so saddles are a particular type of seat with a raised front and back. So all you folks with your brooks' done up too tightly, you've converted your saddle to a seat


Mark (resident pedant)