Q:

What do the terms extraction and strength mean?

A number of questions use the terms extraction and strength in a broad sense like "how much stuff comes out of the coffee beans."
But "technically" speaking: What do the terms extraction and strength really mean for coffee?
Some selected questions that discuss (but don't define as such) strength and extraction:

Coffee package strength markings are often just relative to others within a brand;
Effect of stirring on extraction;
Extracting caffeine and other compounds;
Hot versus cold extraction;
This one and this one about espresso extraction.

A:

Though I don't generally like to cite Wikipedia as my primary reference, it does in this case have a pretty good overview of terms related to coffee extraction, re-summarized here:
Extraction:
The amount of the original mass of the grounds that ends up in the brewed coffee. Though it certainly depends on the preparation method ("method of extraction", e.g., drip, moka, espresso, cold-brew, ...), the recommended extraction amount is about 20% (i.e., about 20% of the mass of the beans/grounds ends up in the brewed coffee). 
Less than ~15%-18% extraction is considered "under-extracted" and can be sour. That is, under-extraction is not just a matter of inefficiency: the result can be sour and unbalanced due to the fact that acids are extracted quickly. On the other hand, more than ~22%-25% extraction is considered "over-extracted" and can be bitter, because the bitter compounds are extracted later in the brewing process.  
These figures were generated by groups with lofty-sounding names like the Specialty Coffee Association of America, the Coffee Brewing Institute, and the Italian Espresso National Institute. Also known as "solubles yield." See also this article on espresso extraction.
Strength:
The concentration of the extracted compounds in the water. Basically, how much coffee is in the water per unit (volume or mass).
As noted in the question, there are also colloquial or vernacular uses of these terms (e.g., "strength" as relative flavour profile; "extraction" as the process of brewing, etc.) but the above seem to be the accepted tangible definitions of the terms.

