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I agree that there is a deplorable lack of agreement as to the definition
of heat.
*) The first definition (heat = thermal energy) is/was used by Carnot
(1824), Boltzmann (1872), Feynman (1963), and every experimental
physicist I've asked about it (including people with PhDs from
California, Cornell, Duke, MIT, and Stanford).
*) The second definition (heat = heatflow) was can be traced as far back as
Slater (1939; Dover reprint 1970) and was advocated by Zemansky in his
1970 _Physics Teacher_ article. This definition has made it into a number
of textbooks and even into the current version of the Encyclopedia Britannica:
http://www.britannica.com/bcom/eb/article/6/0,5716,40566+1,00.html
*) One can dig up additional even-more-unorthodox definitions, but they
are not worth discussing.
Longer-term recommendation: The heatflow definition has got to go.
In my judgement
-- The technical arguments in its favor are invalid.
-- It has no practical advantage when doing thermodynamic calculations.
-- It has no pedagogical advantage, and has serious pedagogical
disadvantages.