I have a question - after reading the assumption clause requirements - when is the assumption clause required? Is it required on refinancing a principal residence?

Regulation Z, §226.18(q) Assumption policy. In a residential mortgage transaction, a statement whether or not a subsequent purchaser of the dwelling from the consumer may be permitted to assume the remaining obligation on its original terms.

Regulation Z, §226.2(a)(24) “Residential mortgage transaction” means a transaction in which a mortgage, deed of trust, purchase money security interest arising under an installment sales contract, or equivalent consensual security interest is created or retained in the consumer's principal dwelling to finance the acquisition or initial construction of that dwelling.

So again, my question is: is this only for the first purchase of a home from another buyer or does it also relate to a refinance?

A refinance does not meet the definition of a residential mortgage transaction.