Yes they're both airspeeds. They correspond to angle of climb, and rate of climb, respectively. Yes, changes in weight, density altitude and configuration (gear and camber of the wing), will alter the value of both. POHs and airplane manuals are written for the lowest common denominator. As such, generalizations and single-value definition are utlilized, patronizing as it may seem to some. I respectfully offer this thread as evidence of why OEMs behave in such ways. Don't shoot the messenger.
As to angle of climb: The cambering of subsonic, moderate to high aspect ratio unswept wings via flap defelction, WILL result in a decrease (that's a lowering, as in bad) of the angle of climb, due to increased drag imparting a cost on excess thrust otherwise available when the wing is clean. People's misunderstanding of the roles of wing lift versus aircraft thrust in climbing flight is why this seems to be a topic of confusion. to take generalizations made by POHs for the sake of simplifying takeoff technique for the lowest common denominator should not be taken as evidence of the contrary.
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Absolutely not.