I, too, find it confusing. The "German cognate is closer" is not helpful!

I think the ö is significant. It could correspond to English ē, but not ei, -ine.

Under sʜᴇᴇɴ, Partridge [1] states that OE scēne, scȳne are related to G schön, from PIE *skauniz "Ultimately, to E sʜᴏᴡ."

I think we have two compartments here:

1. ö/ē words - schön, E shown, shewn. Under Partridge [1] sʜᴇᴇɴ

2. ei words - G schein and E shine. OE scīnan, under Partridge [1] sᴄᴇɴᴇ

[1] My favorite reference: Eric Partridge: _Origins: A Short Etymological Dictionary of English_. More concise than the OED, and you can carry it.

As an English speaker, I'm delighted by the borrowing "ser schön". It is the highest grade in English catalogs of ancient coins. "Shiny" is not a good quality in ancient coins!