programming - When should I use Apply (or Function) and when @@ (or &)?

This is a rather general question, which I fail to answer myself. I guess it is mainly due to my insufficient knowledge of the precise terms.

If I understand correctly, the following are equivalent:

f = Function[u, 3 + u]
f = Function[3 + #]
f = (3+#) &
f[x_]:=3+x

In all cases f[a] would yield 3+a. Similarly, f[a] is equivalent to f@@{a}.

I have mainly two questions:

What is the right term to call the shorter version, i.e. the version where one uses @@, #, & etc.?

When should I prefer the one method and not the other? It seems like many answers given around MA.SE uses the "shorter version". What are the advantages and disadvantages...

Bonus question: Note my first question. I hope I could get more fishing rods and less fish. In other words, what is the official term for these two notions? Where are can I find documentation of these differences? The examples I presented in the this questions are merely examples, and I try to understand where can I rigorously study these notions and similar ones.

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functions - Get leading series expansion term?

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I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...

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