Strength = force required to break a sample.
Toughness = energy required to break a sample.
Because force is not the same as energy, strength is not the same as toughness.
As for tensile, compressive, sheer, impact, etc. these are different modes for breaking a sample. So we have a huge combination of possible quantities, many of which are used by engineers. Here is the Cartesian product of most of the terms possible, followed by some examples
"Failure Mode" x "Force or Energy"
{Tensile, Compressive, Sheer, Impact, Fracture, etc.} x {Strength, Toughness}
Examples include:
Tensile Strength, Tensile Toughnes,
Compressive Strength, Compressive Toughness,
Fracture Toughness, etc.
So I was specifically talking about tensile toughness, which is the energy required to break a sample as load is slowly increased (as you described). Similarly, there is tensile strength, of which there are several kinds. For example, the tension at which the sample begins to deform plastically is the yield strength, and the maximum force over the entire stress-strain curve is the ultimate tensile strength.
Basically, you have a bunch of different ways the sample can break, and you want to analyze both the forces and the energies involved. Based on the material and application, some of these quantities are more relevant than others. So, some things are (almost?) never used, like impact strength.
I'm not a material scientist, but this is my understanding based on what I've read.
Most of my understanding comes from the following sources:
My undergraduate degree in physics, plus:
_Why Things Break_ by Mark Eberhard (2007)
http://www.amazon.com/Why-Things-Br...d=1393260688&sr=8-1&keywords=How+Things+Break
_The New Science of Strong Materials_ by J. E. Gordon (2006)
http://www.amazon.com/Science-Mater...tmm_pap_title_0?ie=UTF8&qid=1393260432&sr=1-3
Wikipedia
Undergraduate lectures such as on Prof. Bhadeshia's YouTube channel:
https://www.youtube.com/channel/UC8gl2omRFMCaStGqiXqNMoQ
Forum members who are material science engineers or students.
Sincerely,
--Lagrangian