I think thinking about it in relation to the English word might be helpful. We also have an underworld connotation to the word "shade." Shades of Stalin! And yet, though that exists, we never make that connection when we talk about the "pleasant shade of a tree."
Now, in slang, kids started using "throwing shade" at someone, i.e. putting them down/insulting them. There too both shadows and ghosts are entirely absent from that usage.
Now, it's possible for a Latin author to make a pun, like your nemus umbrosum example, but it's definitely not necessary to always think there's a pun, especially since meaning (2) is not the primary meaning, but a developed, metaphorical meaning.
What helps solidify disassociating the two is the near (total?) absence of prose uses of umbrae as "ghosts." At least Lewis & Short note that it was poetical and made its way into prose rather late. The closest you get (looking at the OLD now) is Cic. Rab. Post. 41, but that's even closer to the English "shadow" than the dead umbra, in that he's not literally dead.
But if, O judges, you wish to know the truth,—if the liberality of Caius Caesar, which is very great to every one, had not been quite incredible towards my client, we should long since have ceased to have Postumus among us in the forum. He, by himself, took upon himself the burden of many of Postumus's friends; and those responsibilities, which during the prosperity of Postumus many of his friends supported by dividing them, now that he is unfortunate, Caesar supports the whole of. You see, O judges, the shadow and phantom of a Roman knight preserved by the assistance and good faith of one single friend. Nothing can be taken from him except this image of his former dignity, and that Caesar by himself preserves and maintains. And that, even amid his greatest distresses, is still to be attributed to him in an eminent degree.
Livy (40.56.9) appears to be the earliest prose use of it as a synonym for manes.
But he was more sick in mind than in body. He was a prey to gloomy fears and sleeplessness; again and again the form and shade of his innocent murdered son threw him into violent agitation.
All these seems to me to suggest reading the two meanings in the opposite way. It's not so much that a shady place is good for ghosts, but that ghosts are shadowy.