Here is my '62 J.
The body is refin. Neck is original except for frets. Most of the electronics are original except the neck pup which was rewound at Fralin, and where they did not return the rewind on the original bobbin. Long story, but water under the bridge.
I'll say this - determining authenticity of original vs refin can be a slippery slope. For example, I am dead sure my neck finish is original; but the back of the neck is NOT worn off; and it is definitely not a respray. Anyhow, assessing what has and has not been refin'ed is getting harder; and at some point may lose some of its relevance. My < 2c.