Question

What process must occur to define an object as a true star? Are there any objects that are sometimes placed on an H-R diagram that are not true stars?

Answer 1

To define an object as a true star, there must be deuterium (an isotope of hydrogen) fusion in its core into helium and it's it's luminosity should be in enough.

Yes there are objects that are sometimes placed on an H-R diagram that are not true stars. Sometimes super giant planets that have luminosity close to true stars are also placed on H-R diagram. Also the early phase of stars, known as proto-stars, having enough mass and luminosity are also placed on H-R diagram.