Hmmm. I’d say in calcination we’re left with a solid - oxide, carbonate, sulfide etc whereas in pyrolysis (= burning) we’re left with nothing but ash/residue and carbon dioxide/water. I’d argue that most pyrolysis processes are in air.
Calcination is a process of heating a substance under controlled temperature and in a controlled environment... Pyrolysis is extreme burning of organic materials (usually)
These words are used differently in different parts of the industry, so be careful who you talk to. Usually, pyrolysis decomposes substances beyond their chemical stability under reducing conditions. Calcination can be a simple annealing of the material.