For any international tournament, correct, it is that simple. Many domestic teams - pro, school, university, etc. - have non-Victor sponsors and the players use their racquets and/or attire for domestic tournaments.
that's universally true. domestic school teams are independent of the national team. we are usually referring to the national team in these discussions.
Endo/Hayakawa would be another example. They both use the typical Japanese Yonex clothes and Mizuno shoes, but Endo uses Yonex rackets while Hayakawa is apparently sponsored by Kumpoo (even though he never puts a logo on the racket).
Also, Tan/Blair - I'm curious, does anybody know what Blair's sponsorship situation is? He was wearing an Apacs shirt, but everything else was Yonex (without a stencil on the racket). Is the pairing sponsored by apacs, but Blair individually endorse by Yonex? Can't imagine he buys his equipment (as a pro, that'd be rather expensive as he probably goes through one racket a month at the least)...
Blair is sponsored by Yonex as far as equipment goes. Free equipment. Not sure about Apacs connection
Perhaps some kind of individual deal, wearing the shirt in competition with TBS might get them a salary or bonus for reaching latter stages of a tournament...if Blair didn't get anything out of it, he probably wouldn't wear it, as I'm sure he could get an orange Yonex shirt to match Tan's