That FAQ also says "If you have a US bank account, you may be able to transfer funds to a Bangkok Bank account in Thailand via your bank’s internet banking service, using Bangkok Bank’s New York branch’s 9-digit routing number."
If so, that would seem to be cheaper. I don't pay fee for international wire from my Citibank account, but Citibank exchange rate for transfer (33,3667 at moment) is bad whereas I assume if go via Bangkok Bank NY it would be at more favorable Bangkok Bank rate.
I wonder if in addition to using that routing number, you need to tell them to deposit into your account in Bangkok -- besides, of course, giving them your account number.