No. A cashier's check must be issued by and drawn on the same bank. The check you've described is drawn by a bank depositor, not by the bank itself.
If the entity issuing check is a bank, it's either a cashier's check or a teller's check. If the entity issuing the check is not itself a bank, it's not a cashier's check or teller's check.
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John S. Burnett
BankersOnline.com
Fighting for Compliance since 1976
Bankers' Threads User #8