There’s a difference between nicknames and diminutives. Diminutives are variations on a name that are based upon the sounds in that name. Bill, Billy, Will, etc. are diminutives for William. Calling the same person “Red” because his hair is read is using a nickname. Calling him “hotheaded Bill” is technically both
Using the name Kate, Kat, or Kathy in reference to someone named Katherine is also using a diminutive. Referring to my low-German ancestor Trientje by the name Katherine is not using a nickname or a diminutive. In this case it is a anglicization (translation into English) of the name Trientje. Your non-English speaking ancestors “new names” may not have been nicknames or diminutives at all–they could have been translations of their actual name.