Conventions for describing tensors and tensor networks:
The range of values a single index can take is its dimension.
The number of indices of a tensor is its order (a matrix is an order-2 tensor).
The tensors making up a tensor network are factors or factor tensors.
For a given tensor network, the contracted indices between factor tensors are internal indices. The uncontracted indices corresponding to the indices of the tensor the network represents are external indices.
The term rank refers to the minimum dimension of a factorization of a tensor with respect to some bipartition of its indices.