We think of "western" civilization of having its origins in the ideas and culture of Greece and we don't think it strange to see buildings using design motifs or even Greek deities from Greek temples -- just look at Washington DC. We use these on our buildings because we admired Greek culture and philosophy. Greece did not directly influence or conquer us.
The term "influence" implies dominance or inferiority on the part of the adoptee so we use the term "Localization" instead -- an awkward term, but they haven't found a better one yet. The presence of a Hindu deity in Java is similar. The Javanese "localized" Indian religion and culture.
By the way, I gave an 8 minute presentation on this piece this afternoon.
Blogs I follow