Geometric entropy vs entanglement entropy (dependent on curvature coupling parameter)

Question

I have a quick question. In hep-th/9506066, Larsen and Wilczek calculated the geometric entropy (which I believe is just another name for entanglement entropy) for a non-minimally coupled scalar field in Eq. 3.10 of their paper. They basically calculated the Heat Kernel on a cone with a non-zero curvature coupling parameter $\xi$ and then found the corresponding appropriate effective action and from there geometric entropy. They also discussed problems about calling this entropy geometric for its possibility of being negative etc. But still they kind of try to explain why geometric entropy can in fact depend on background curvature and so on (the rest of their section 3.3).

But for example Solodukhin (and also other places) in 1104.3712 clearly states the (similar) problems of calling it entanglement entropy near his equation (144).

So, my question was whether it is a settled matter that calling this type of entropy (one which depends on non-minimal coupling parameter in general), entanglement entropy is wrong. Please let me know if there are some subtlety involved that I am missing. Thanks in advance.

This post has been migrated from (A51.SE)