How can I distinguish between elipsis type 1, 2 and 3, below? Type 1 and type 2 "nice day" and "sleeping dog" are both NPs. Type 3 "very sexy" in an AdjP.
Can I say that Type 1 and Type 2 are "NP elipsis" and Type 3 is AdjP elipsis (even if it wasn't the NP/AdjP that was elided?)
Or is there a better way?
It seems to me that you are asking whether what matters is what you start out with or what you wind up with. Broadly speaking, transformational/generative grammar counts what you start with (you could say it's "speaker oriented"), while non-transformational phrase structure grammar counts what you wind up with (you could say it's "hearer oriented").
Your examples look, at first sight, as if what counts is what you wind up with, since in all the cases, you're left with a single constituent.
As a matter of terminology, it would not be wise to say that an ellipsis which removes everything except for a NP as NP-ellipsis, because people would think you meant that a NP was elided, not that it was left behind.
Your examples involve a type of ellipsis mechanism that is not yet widely acknowledged. It has been called "left edge ellipsis". This ellipsis mechanism reaches in from the left edge of the utterance. Your examples do NOT involve any of the more widely acknowledged types of ellipsis (NP ellipsis, VP-ellipsis, gapping, stripping, sluicing, answer fragments, etc.).
Left edge ellipsis has been explored to some extent in works by Napoli (1982), Wilder (1997), Merchant (2004), and a couple of others.
Here are a few more examples of left edge ellipsis:
(1) a. [Do] You want coffee or tea?
b. [Do you] Want coffee or tea?
c. [Do you want] Coffee or tea?
(2) a. [That is] Ridiculous!
(3) a. [I] Have been working a lot lately.
b. [I have] Been working a lot lately.
c. [I have been] Working a lot lately.
(4) a. Pointing: [Pass the] Salt!
What is elided in cases of left edge ellipsis is pragmatically determined by the context in which it occurs. Conventionalized situations give rise to frequent occurrence of this mechanism.
A caveat: theories of ellipsis vary. Some researchers who have studied sentence fragments like the ones in the question would claim that ellipsis has not occurred in such cases at all. They would deny the existence of the mechanism that I, following Wilder (1997), have called "left edge ellipsis".
Finally, I agree with Greg Lee's comment about how the terminology is used. Ellipsis mechanism tend to be named according to the category that is omitted, e.g. N-ellipsis elides a noun, VP-ellipsis elides a VP, etc.
