Why is short-range entanglement defined in terms of its possible deformations?

After reading the question and answers in Definition of short range entanglement I wonder why the definition of a short-range entangled state is given in terms of its possible deformations -

A SRE state is a gapped state that can be smoothly deformed into the trivial product state without a phase transition (all the symmetries are allowed to break during the deformation)

Shouldn't the definition be more like "An SRE state is a state where entangled degrees of freedom are found at a short distance away from each other"? Does this property characterize SRE states? Is it well defined (or basis dependent?)