i've worn buttoned shirts for many years and to be honest, this has never crossed my mind. do we call it a "button up" or "button down"? for the select few (everyone) that already knew the difference between the two, forgive me. but for those that are learning this for the first time, you not alone.
a "button up" shirt does not have a button on the collar. this allows the collar to sometimes flare out or rest spread wide.
i usually stick to button ups for any formal events. wearing these with a tie, helps tame the collar from flaring out too much.
(aside from formal, linen button ups are also a good choice for a relaxing loose fit)
a "button down" has a button on the collar. this keeps the collar from flapping and maintains a more structured look.
normally, i'll wear button down shirts for casual events. with or without a blazer, both looks work well.
both are good shirts that can be mixed in with a variety of outfits, so the next time you come across one, you'll know exactly what they're called and how to utilize them into your style.