I am currently a graduate student and a member of an organization in my field. In my organization, there is this Early-career group composed of young scientists who already earned their Ph.D.'s, taking post-docs, and running their own research groups. In the case of graduate students, does early-career or the term applicable? and what is the difference between the terms early-career scientist, a budding scientist, young-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?