The line between colonizing and mere dominating gets blurred.
That's the point.
India was definitely a colony of the British Empire,
After it was a territory owned by a publicly held company.
since Indians were never even nominally represented at Westminster and were governed by the Raj acting under the Colonial Office and not the Home Office.
Well, none of the conquests of the Muscovite state (Great Princedom/Tsardom/Empire) was nominally represented anywhere until 1905 just due to the absence of any representative body. Does this mean that the Baltic Provinces and Ukraine had been Russian colonies?
Then, when the US had been expanding, does this approach mean that all the new territories which did not, yet, made it into the states were "American colonies" just because they were not represented in Congress? Should we call Puerto Rico a colony instead of "unincorporated territory"?
You can't convincingly extend a British-based model to the rest of the world. Absence of a "Colonial Office" is not a good criteria either because the colony can be ruled by viceroy or governor directly subordinated to the central government.