TITLE: Allowing $G$-CW complexes to have more general cells
QUESTION [18 upvotes]: Let $G$ a finite group. I've seen three options discussed for making $G$-cell complexes: in increasing generality, one might allow $X_n$ to be constructed from $X_{n-1}$ by attaching cells of the form 

$G/H\times D^n$, where $D^n$ has trivial $G$-action, or
$G/H\times D(V)$, where $D(V)$ is the unit disk of a $G$-representation $V$, or
$G\times_H D(V)$, where $D(V)$ is the unit disk of an $H$-representation $V$

(The subgroup $H$ and representation $V$ is allowed to vary for different cells. Cells of type 1 are what are used in the standard definition of $G$-CW complex.)

Megan Shulman, on p.47 of her thesis (link goes directly there), says regarding standard $G$-CW complexes that 

If we are interested in putting CW structures on spaces found in nature, this is a very restrictive definition...

However, I seem to remember that even for a $G$-cell complex $X$ constructed with cells of type 3, we can always rearrange (triangulate?)  somehow to give $X$ a standard $G$-CW complex structure. Is that true or false? Which papers look into this?

Ferland and Lewis, on p.23 of their paper (link goes directly there), say that cells of type 3 

... are of interest because they arise naturally from equivariant Morse theory (see, for example, [21]). Further, if $G$ is a finite abelian
  group, then the usual Schubert cell structure on Grassmannian manifolds generalizes in an obvious way to a generalized $G$-cell structure on the Grassmannian manifold $G(V, k)$ of $k$-planes in some $G$-representation $V$ (see Chapter 7).
[21] A. G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150.

However, skimming over at that section of Wasserman's paper (link goes directly there), I must admit I don't see where anything like cells of type 3 show up, much less demonstrate their usefulness. Could anyone explain their relationship to equivariant Morse theory?
Also, in their Chapter 7 (link goes directly there; look at the bottom of p.83 & top of p.84 in particular), I'm not sure I see where cells of type 3 are necessary, because it seems like $W$ is inherently a $G$-representation, and hence they are only obtaining cells of type 2. Could anyone clarify this?

So to summarize, I'd greatly appreciate any of the following:

simple / natural examples that illustrate the additional flexibility of each generalized cell:

spaces which can easily be seen to have a cell structure when cells of type 3 are allowed, but not when only cells of type 2 are allowed
spaces which can easily be seen to have a cell structure when cells of type 2 are allowed, but not when only cells of type 1 are allowed

any other reasons why cells of type 2 and 3 are reasonable
a citation / proof / counterexample for this half-remembered "fact" that cells of type 1 suffice

REPLY [3 votes]: Just a quick note adding to Steve's answer, with a focus on the stable setting. In the Hill-Hopkins-Ravenel paper on the Kervaire problem, they introduce the "positive complete model structure" on G-spectra, and the "complete" part gives you cells of type 3. This is on page 161 of the last arxiv version (which I think is super close to the published annals version) https://arxiv.org/pdf/0908.3724v4.pdf
They needed these cells in many places in the paper, but most notably to prove that, for every cofibrant spectrum $X$, the natural map $(E\Sigma_n)_+ \wedge X^{\wedge n} \to X^{\wedge n}/\Sigma_n$ is a weak equivalence (where $\Sigma_n$ denotes the symmetric group on $n$ letters). They use this property to set up a good homotopy theory of commutative monoids. This property was claimed without proof in earlier sources setting up the homotopy theory of equivariant spectra, but the proof is really non-trivial. It follows for the positive and positive flat model structures, but I only know how to prove it using the positive complete model structure. I believe Doug Ravenel has continued to investigate this model structure. I guess my point is that you should not restrict yourself to looking at classical sources, because the decision about which cells to use was still in flux till very recently. Ravenel gave a great talk a few times in 2015 where he argued that this positive complete model structure is the definitive right answer (at least, for equivariant stable homotopy theory).