In the end "strength," in the non-technical sense, is a competition.
The endgame is to produce a feat unlike what others have done before you.
A lot of people can bust out 250x20, but how many squat a grand?
It's about doing something, and hence becoming, special.
It's about compiling all you're work, and the work of your ancestors, into one original moment.
In the technical sense, strength is the ability to produce force, absorb force, or withstand force, which is kind of running the gamut of athletic achievement. That's why there's a bunch of different athletes, with different needs and goals, at different levels of development who can all be considered "strong." It's all about the attainment of the goal you set out to achieve.
For instance, Mariusz Pudzianowski is strong, as is Dmitri Klokov. One a world strongman, the other a weightlifter.
Two extremely different builds, with completely different movement skills, both accomplishing unbelievable feats in their special area of "strength." We can go even farther, compare Bruce Lee and, oh, how about Vincent Wilfork? There's no way that either of those men are considered anything less than damn strong, but look at how different they are, their goals and their genes define who they are.
So really, "strong," is whatever you want it to be, and whatever your definition, it will decide how you train, and what you get in terms of results.