Definitely not. These leaves are much broader and lighter green than Pouteria caimito, and the shape is obovate-oblong; P. caimito has darker green, more lanceolate leaves with slightly wavy or "scruffy" margins. I'm not even sure that it is a Pouteria, but the spiral leaf arrangement and the "caimito" name makes that seem like the most probable genus.
As far as I know, Pouteria caimito is called "abiu" or "avio" everywhere in Ecuador, though the indigenous tribes probably all have their own names for it as well. The fruits commonly called "caimito" are Chrysophyllum cainito and Chrysophyllum venezuelanense, but that genus does not have the spirally-arranged leaves. Is everything a "caimito" now? I remember when everything was a custard apple...