Why speed was defined as distance over time ? Are there possible ways to define speed like distance plus time or distance minus time or distance times time
If you think about what we mean when we say velocity, it is how much the position changes over some period of time. This is clearly given by $v=s/t$. Move double the distance in the same time, you have double the speed. Or the same distance in half the time, also double the speed.
Distance plus or minus time makes no sense, as the units don't match. Distance times time also does not have the properties we want if we ask how quickly something going from $A$ to $B$, and has no significance in physics that I'm aware of.