I get confused knowing which bin to put things in. The bins at Wholefoods are nice in that you get your food from them using their packaging and then the bins have photos of which items go into which bin. Better communication could increase circular economy as it would avoid people putting items in the general bin due to confusion. How many people are willing to admit they are confused?
On your small changes, writing on lid. "Don't take a lid unless you really need it." could reduce how many people use them by a great amount and would save costs for coffee shops. I have never been asked whether I really need the lid and get them more out of habit than a necessity on certain occasions.
Or maybe they should be different type of lids. The plastic lid is for drinking on the go. It is useful for people drinking their coffee while driving or walking. But if someone is buying coffee and bringing it back to the office, maybe a different more environmentally friendly lid could be used that doesn't need to allow people to sip it, no need for an opening and therefore can be cardboard or paper. Or if coffee shops sold jugs of coffee for an office worker to purchase a jug for four rather than four individual coffees, would that save on packaging and be useful for both coffee shop and consumer.