Well, there are various ways through which cognitive abilities can be assessed. Mathematical ability, perception, visuospatial abilities, verbal reasoning, and problem-solving ability are among the general consensus, just to mention a few. I do not think any researcher will just impose their own perception of what they think constitute cognitive abilities without recourse to literature.

I would be curious to know if there is research looking at the different ways left-handed and right-handed people go about doing certain things.

I think some examples were given in my write-up unless you are talking about physical strength as opposed to cognitive ability. Also, here is another example http://ijcrr.com/uploads/1218_pdf.pdf