There's not really any confusion in wording.
Let's be clear here. By providing Displaylink the permission to Record/view the screen, that's exactly the permissions and data being made available to it.
In other words, it's giving closed source propriety software product full access to your screen data and everything that appears on it. What does this 'black-box' software do with that data? We don't know, because it's closed source and propriety.
Yes we know it needs the permission to provide the functionality it does - which is a workaround way of sending video data via USB (because Apple failed to provide basic dual-external monitor functionality), but can we be sure what else DisplayLink does with that data? No.
Is it safe? We can't be certain.
Is it recording/saving/keeping/doing analytics on all of the data that appears on your screen including sensitive, private and potentially incriminating information? Is it selling your data? Training AI with it? Maybe - we don't know.