Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category?
if we don't assume properness, I don't even see why the first is homotopy-invariant!
The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration is always its homotopy pushout in any model category, see Proposition A.2.4.4 in Lurie's Higher Topos Theory.
This is the case for both of your examples, since the initial object is cofibrant.
An argument showing that the two models of suspension are equivalent will probably be based on something like the following:
Assertion: Suppose we are given a commutative diagram of the form $\require{AMScd}$ \begin{CD} \ast @<<< C @= C \\ @VVV @VVV @VV V \\ Y @<<< A @>g>> X \\ @| @VVV @VVV\\ Y @<<< A/C @>>h > X/C \end{CD} in which the vertical directions form cofibration sequences (when I write $A/C$, I mean $A \amalg_C \ast$, where $\ast$ is the zero object), and the maps $g$ and $h$ are cofibrations.
Then the map of pushouts $$ Y \cup_A X \to Y \cup_{A/C} X/C $$ is a weak equivalence, or better still, it is an isomorphism.
It seems to me that this is true by the assumption of properness, since we have a cofibration sequence given by the pushouts $$ \ast\cup_C C \to Y \cup_A X \to Y \cup_{A/C} X/C $$ in which the first term is isomorphic to $\ast$.
Let's call the first suspension $SX$ and the second one $\Sigma X$.
Given the assertion, we can show that the two models for suspension are weakly equivalent as follows:
Apply the assertion to the diagram \begin{CD} \ast @<<< \ast\amalg X @= X \\ @VVV @VVV @VVV \\ \ast @<<< X\amalg X @>g >> \text{Cyl}(X) \\ @| @VVV @VVV\\ \ast @<<< X @>>h > CX \end{CD} (where $CX = \text{Cyl}(X)/X$) to get that the map $$ SX\to \Sigma X $$ is a weak equivalence.
If you're looking to learn more about homotopy colimits, I strongly recommend:
- Dugger's Primer on Homotopy colimits
- Shulman's Homotopy limits and colimits and enriched homotopy theory
- Rehmeyer's 1997 master's thesis (under Mike Hopkins), "Homotopy Colimits"
- Homotopy Limit Functors on Model Categories and Homotopical Categories by Dwyer, Hirschhorn, Kan, Smith
- Riehl's book Categorical Homotopy Theory
I note that the first four predate Lurie's books, and the fifth works out many examples. The fact that the pushout and homotopy pushout agree for a span diagram when all objects are cofibrant and one leg is a cofibration (even without left properness) is 13.10 in Dugger's manuscript. A detailed treatment of the cofiber is in Rehmeyer's thesis. Shulman handles your other question, about why these two ways of computing the homotopy colimit agree (e.g., Section 5, drawing on Dwyer, Hirschhorn, Kan, Smith).