Skinny
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Those are very generic statements. Typically a lean mixture will run hotter EGT's where rich mixtures actually burn cooler than lean ones. Small carb's piston aircraft have mixture adjustemets where a pilot changes the mixture based on EGT and rpm.Diesels have cooler exhaust temperatures, part of the high compression ratio and the fact heat is work is energy. Greater compression ratio, cooler exhaust. That and the fact diesels usually run far lean.
Now in a turbodiesel, the EGT goes up with more fuel only because you are feeding it more combined air and fuel which simply creates more heat than an engine at lower boost levels. Hence the EGT increasing with a rich mixture.