How do we identify a Scientist as Early-career scientist?

"Early-career" can mean any number of things, depending on who you ask and in what context (from just PhD students, to everybody except tenured professors). Usually, it is just a label and hence matters little.

Where it actually can and does matter is in the context of grant applications, as many funding agencies have special calls for early-career researchers. However, in these cases, the agency will specify exactly what the boundaries of eligibility are. For instance, the Swedish Research Council (VR) has a Startup Grant that is available between 2 years and 7 years after getting a PhD.

Exactly what any particular organization means by "early-career" is determined by the organization. The main point is that the term "young" is condescending and ageist. If you get a PhD at 50, are you a "young" scientist?