Be/keep/stay in touch

To me all three mean different things.

Let's be in touch I say this to someone I've never before met, but during the course of our conversation an important matter emerges which affects us both. It seems important that we maintain contact.

Let's keep in touch I say this to a colleague with whom I have worked for several years, but our future paths are about to diverge.

Let's stay in touch I have just attended a conference of people working in the same industry. many of the participants were new faces to me, some were old friends. At the end we all appreciate that we have gained a lot from the meeting, and someone says 'let's stay in touch'

I have exaggerated the differences here and any of the three could easily be interchanged with either of the others. But the beauty of English is that there are always multiple ways of saying almost the same thing. So this gives rise to a potential richness, and possibilities for finely nuanced expression. One can bring joy to one's fellows through sweet words; as long as one never forgets that 'sweet words butter no parsnips'.

All correct, but mean different things:

Let's be in touch: Call me later to discuss this [after an event has occurred].

Let's keep in touch: I hope to keep hearing from you in the future [now that we've reconnected with each other].

Let's stay in touch: May mean the same as keep in touch, but also may mean, Check back with me periodically to discuss this.