It is best not to think of imperialism as a single event committed by one polity upon another. Imperialism can be overlapping, and an imperializing power might be being imperialized by another. Often, imperialism is used to mostly indicate a form of colonialism conducted by European powers. However, you are correct that non-European imperialism existed/exists. In Africa, Ethiopia, the only African power not to be colonized, can be read as an African-grown imperial state (albeit with less global reach than comparative European imperialists [A History of Ethiopia by Marcus]).

The definition debates surrounding imperialism are a confusing mess. Edward Said has a particular take on it, as does World Systems Theory, Hobson, Kautsky, Lenin, Luxemburg, and others. Many of these definitions seem to vary only slightly, but in actuality have huge philosophical ramifications. Since the publication of Capital in the 21st Century, Lenin's definition and approach is taken more seriously than previously at least from a historical economics standpoint (even if you disagree you still read it), provided one can get past the outdated teleological undercurrent. How imperialism differs from colonialism is not always clear, but the general sense from these various opinions is that it tends to be economic-centric domination, although this has been disputed (especially as the usual corollary is to emphasize colonialism in contrast as settler-driven, which presents problems and perhaps-dogmatically juxtaposes economic concerns and settler concerns). Therefore, whether the relationship between Bahrain and Iran would be imperialistic would depend on how one defines imperialism.