I can confirm the former. Any character chained with the initiators will take into account the type of the initiator and it's own stats. As for skills, I do believe it is of the same mechanics; regardless of the actual type of skill used, only the type of the initiator is actually taken into account. I can't say for sure for skills as I have not closely observed this enough, but this is what I believe to be the case...
Example: A multi-type (both lance and bow skills available for use) skilled sword character initiates a pincer against a lance enemy. The lance and bow skills do not take into account its' type when doing damage, thus all skills regardless of type will do the same damage.