Dog shelter – used as an umbrella term to refer to any type of animal control, shelter, or rescue/rehoming organization; it is acknowledged that the breadth of such organizations is wide and can vary considerably in terms of several factors (e.g. municipal versus private, multiple branches versus single location, dogs housed in a dedicated facility versus individual foster homes)

Shelter dog – refers to any dog that is part of any type of animal control, shelter, or rescue/rehoming organization 

Dog rehoming – refers to the act of finding and placing a dog in a new home

 Adopt and rehome [a dog] –  refers to the act of acquiring a dog from a shelter/rehoming organization (the latter tends to be used less commonly in the US)

Relinquishment [of a dog] – refers to the act of surrendering an owned dog to shelter/rehoming organization, regardless of the origins of the dog or from where the dog was acquired