This started as a comment but got too long.
I think you have some good answers already regarding the naming of precipitation hardening, so my answer attempts to address some of the other semantic issues it appears you are struggling with.
Area under the curve is term which refers to a stress-strain curve, which (as it sounds) is a graphical representation of a materials behavior under stress. This is a measure of the materials strength, and does not necessarily indicate toughness, or hardness.
Strength is a materials ability to resist a load (known as stress). When a load is imparted to a material, it will undergo a deformation. The measured amount of deformation is called strain.
A stress-strain curve therefore is simply a plot of the amount of strain (deformation of the material) corresponding to various levels of stress (load). Typically this is plotted with strain along the horizontal axis and stress along the vertical. There are different ways these curves may be expressed e.g. engineering strain vs true strain, but I won't go into that here. I mention it as something to be aware of if you're going to be reading and comparing curves.
Now, I mentioned this is a measure of material's strength, and I'll attempt to give a brief explanation of this here. For metallic materials, a stress-strain curve is typically generated from a uniaxial tensile test, which measures the materials ability to withstand an ever increasing load in one direction under tension. The standard test consists of rigging a (typically cylindrical) sample of material between two clamps, one clamp is stationary, and the other is attached to a platform which moves away from the stationary jaw. This movement apart applies tension to the sample. This tensile load is increased until the material breaks, and the highest load achieved before rupture is known as the ultimate tensile strength of the material. (So UTS refers to the maximum amount of tensile stress a material can withstand before failure)
Aside from ultimate tensile strength (UTS) the other common measure of strength you'll see is yield strength (YS).
With metallic materials there are typically two distinct regions in the stress-strain curve; elastic and plastic. As load is increased from 0 up until the yield point the strain will increase proportionally at a linear rate relative to stress. At loads (stress) of this intensity the material behaves elastically, which is to say that when the load is removed, the material will return back to its original shape (stress=0, strain=0). At some point, the stress will become great enough to overcome the material's elastic limit, meaning permanent deformation occurs (stress=0, strain>0). This permanent deformation is known as plastic deformation, and so the area under the curve in this region is known as the plastic region.
The point at which this transition occurs is known as the material's yield point. Typically, it's not distinct enough to measure the exact point at which yield occurs, so an approximation known as offset yield is typically used. You'll commonly see material specifications refer to 0.2% offset yield (or some other percent offset) which refers to the approximated yield point. I won't get into the methods for offsetting to determine approximate yield, but it's easy enough to google if you're interested.
One more thing regarding material strength; so far we've only discussed tensile strength, which is the most common measurement of metallic materials, however other measurements of strength exist as well. For example, ceramics and cements are not commonly measured under tension but rather under compression, which is known as compressive strength. This again will be graphed as a stress-strain curve, but is a different test involving compression of the material until failure. I just thought it worth mentioning as it relates to the understanding of the word "strength" in materials science.
On to hardness.
Hardness is typically measured as the materials ability to resist indentation. Various standards exist for testing hardness, such as Rockwell, Vickers, brinnell (all common with metallics) and shore (common with rubbers and plastics).
While hardness is measured as resistance to indentation, it's commonly used to judge a materials resistance to abrasion and wear, and can can be manipulated to increase a materials performance in certain applications or to improve machinability/workability of the material.
Hardness and strength commonly have a positive correlation as the mechanisms which increase a material's hardness typically also increase the material's strength.
Toughness typically has a negative correlation to hardness. Toughness is a measure of a material's ability to resist fracture under load. Brittleness is the opposite of toughness, and typically as a material becomes more hard, it also becomes more brittle. This is not the rule, just typical for many metallics.
Toughness is a little more difficult to quantify than strength, as there are many ways for a material to to be loaded to failure and for fracture to occur. Some common indicators of toughness are impact resistance (charpy is a common standard test), fracture resistance/resistance to crack growth (plain-strain fracture toughness is a common test), and ductility.
Ductility is similar to toughness, but is not necessarily related to a materials ability to resist fracture, but is rather a measurement of its ability to flow and deform under load. Ductility is commonly expressed as reduction of area or percent elongation; both are measures of plastic deformation before failure, typically established from the tensile test.