by Jim Bedsole:
The term of the loan is a triggering term under Reg Z. Unlike deposits, where Reg DD only has the APY as a trigger term, Reg Z has several possible trigger terms, any of which require all of the disclosures you mention. See 1026.16(b) and 1026.24(d) for listing. The only clarification I'll make is that down payment is only a requirement as relates to credit sales transactions - those where the creditor is also the one selling the goods or property being financed.
Both of the advertising examples you list do include trigger terms and require disclosure of the additional terms.
by Richard Insley:
The cite for this is the Official Interpretation of Section 1026.24(d)(1) and definitional Section 1026.2(a)(18). The term "downpayment" is defined exclusively to be an element of a credit sale transaction.