What are the names of the conventions for defining the double dot product?

Question

The double dot product of two matrices $A : B$ can be defined as either:

1. $A : B = Tr(AB^T) = A_{ij}B_{ij}$
2. $A : B = Tr(AB) = A_{ij}B_{ji}$.

I've seen the first convention called Frobenius or Holzapfel and the second one called Reddy.

Do these conventions have widely accepted names ?