You basicly already arranged it fairly good: Geliebt < sehr geliebt < innig geliebt < heissgeliebt
"Geliebt" just is beloved. "Sehr geliebt" is literal translated very beloved. Both can be used in a wide range of situations beginning with e.g. use in a laudation up to romantic love.
"Innig geliebt" is already very strong. I think its only used for all kinds of true love. For romantic love but also e.g. how the father loves his child.
"Heissgeliebt" means something like loved with passion (Heiss means hot). It's used for romantic love but not e.g. for the love a father has for his child. This word also has a second meaning which could be translated as "he/she really really loves x", x could be chocolate.
If your parents have named you Amy also because of the meaning of the name I would use "innig geliebt" as translation since it seems they really wanted to take a emphasis on the love they have for there child.
You also could go with the german translation of the latin word amatus on which the english name Amy is based. That would be "geliebt", but it dosn't sounds as nice as "innig geliebt" and I believe a more free translation is okay here ;-)