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In Symposium: the Road to Reality (22.01.2019), part II - Sir Roger Penrose a few diagrams are presented that include a "Physical World", a "Mental World", and a "Platonic/Mathematical World".

What definition is available for what is meant by "world" here?

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Penrose is using the three world model of Frege and Popper. Here is a link to Popper's Tanner lecture which explains three worlds. https://tannerlectures.utah.edu/_resources/documents/a-to-z/p/popper80.pdf

In summary, there are three kinds of things in our universe -- things which have location and time properties, and those things are basically matter, and are world 1. Things which have no location, but do have time properties, those are conscious experiences, and they constitute world 2. And there are things with no location nor time properties, these are abstract objects, and they are world 3.

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