Yeah, getting to a good definition of UX vs. UI that everyone could agree on would be great ;)

I'd start with UX defined as "what the users actually ends up doing while using your UI, the actually followed user journeys, the workflows they lead to, all this from the user's subjective perspective (hence the 'experience' part)". Also relating all this to goals and talking about "conversion" etc. would be about UX. And the general feeling of "this really helped me get the job done" that the user gets after using something. Heck, even customer-service can be part of UX if your product does something so complex that having users regularly and often interact with customer service is part of the "regular usage pattern" (think stuffs where CS is not just where you complain, but where you may have talks with reps about "special shipping and assembly arrangements" etc.)

And UI would be "what and how can the user possibly do with your interface, what are the actual interface elements that actually dictate these "how" and "what" of the user's actions, things like responsiveness, intuitiveness (matching user's previous expectations), information density, discoverability". So you can talk about "user satisfaction/enjoyment" here, about "intuitiveness" etc. (but these don't necessarily mean 'higher conversion' or making the user more productive!) Also accessibility comes here. And think things like "is this button big enough to be tapable" or "it that option in hover-only menu that is impossible to access on mobile devices" or "is that other thing a keyboard only shortcut that tablet users will likely miss" etc. These are obviously independent form how things look: the design can be horrible, but everything can be easy to find/tap/click etc.

Graphic design is obviously "how things look" and this also affect the "how easy is to find things", to it does affect both UX and UI... but, there is a big "but" here: designs that receive awwards and are perceived as beautiful tend to have functionality harder to find, and more confusing user journeys. Think about the "ghost buttons" that people used and still use... but have been proven to decrease conversion quite a lot! ...but now the trend might have reverse because users are finally getting used to them. Like, do you really want to be playing this fashion game that can hurt conversions when you don't expect it in a resource constrained startup?! If you're Apple, you can "reeducate users' aesthetic tastes" and stuff which is cool, but you're not Apple. So better under-design a bit just to be safe :)