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Sometimes in correspondence with faculty I get emails that end with a sentence like "happy to find a time some time to chat more about it though, just let me know."
What medium do people usually mean when they say this? Is this an invitation to schedule a phone call? Or because this is on an email is it best to stick to to sending responding with a detailed email about the said topic?
The "just let me know part" invites you to let them know your preferred times and propose a way to have a chat.
The typical medium to let them know would be via email. And then the typical medium to chat would be in person. But that's it's not so important. You've been invited to make a suggestion without much advice. So any response that is both reasonable and polite should work. Even if the professor will "counter" your suggestion with an alternative.
Send him a meeting invite. If your topic is small, just ask for 15 minutes. If long, ask for 30 or 60 or whatever. A good time to ask for is 5PM. Copy his assistant, if he has one. If he is not available, he will just counter. But just take action and send the actual invite.
Face to face is best (location "your office"), phone is next best (or even better Skype/Zoom, etc. now that even the geezers has been forced to learn them). But phone is still fine, especially if he doesn't want to do video. Don't ask for a chat meeting...that is bizarre and distracting. IF he volunteers that, then fine, take it.
You don't need a document, but have a simple outline (can be in the meeting invite description field) for breaking down the topics and making sure key questions are asked. (You can allow the meeting to be natural and back and forth, but having an outline will help make sure the time is not wasted.)
Given the modern age, I doubt that it is an invitation for a phone call. An office visit perhaps, or a text back and forth via email or a chat app. I would probably be offering an office conversation myself, but others might not. But the phone is a poor mechanism for such things. It requires an immediate response to questions and offers poor opportunities for capturing important things.
At this moment, of course, April 2020, face to face meetings are generally impossible, but in normal times that would be my basic assumption. For a department with a coffee lounge it is pretty natural to meet there and discuss whatever is of interest. Normal times. Hope for their return.
But a detailed email might be overkill. Think of it as more of a prolonged conversation and don't flood the recipient or you may wait a long time for a reply if they are busy.
Is this an invitation to schedule a phone call?
Yes, or an in person visit, or a video call. I would describe this sort of statement as extremely generic and carrying little meaning. It's like saying "Have a nice day."
When I write something like this in an email, the primary purpose is to signal that I am open to continuing discussion of the topic, and my response shouldn't be seen as the final word. Sometimes students (and others!) are shy about coming back for clarification or to point out flaws in my response to their question, and I want to encourage them to do so.
Generally, unless you're someone I know well, I wouldn't expect a phone call or in-person visit as an immediate response. Either continue the discussion by email, or send an email proposing that the matter is best resolved by some other medium (phone/video call/in-person meeting/etc) and asking to schedule a convenient appointment.
