In this paper there is a clear distinction between steepest descent method and saddle-point approximation method. Eg. in the first page it says:
Indeed, Picard-Lefschetz theory establishes that a saddle point can contribute to the path integral only if the saddle point can be approached from the original contour along a steepest descent contour.
There are other examples also. Am I misunderstanding the statement? Aren’t the two methods the same?
Brief summary:
The steepest descent method and the saddle-point approximation method are the same.
The stationary phase approximation is the special case when the integrand is a phase.
Concerning OP's quote, note that generically only a subset of the saddle points contributes to the integral in the asymptotic limit.
For an example, see e.g. my Math.SE answer here.
Laplace's method is for a real integrand and without contour deformation.
In the WKB approximation, the starting point is a wavefunction Ansatz for a differential equation rather than an integral.
Qmechanic's answer is just too good.
I'd like to add that both Laplace's method and stationary phase approximations are integrals along the real number line, covering two opposite cases, and are today correctly taught as if they were lemmas on the way to the steepest descent method, which is also known as saddle point method.
The crucial addition is the contour deformation: from the original contour, nudge the contour to reduce the integral to either the stationary phase approximation case, or much much more commonly, to the Laplace's method case. This latter case is why the method is called steepest descent, and tells us what to do when we encounter a saddle point. It also makes it clear that only the saddle points reached by the appropriate nudging of the contour from the original, would contribute. Saddle points that are not along the nudged contour do not contribute.
Needless to say, the lemma-like presentation of these topics encourages students to conflate them, that all of these things are grouped together as steepest descent, not least since the main result obtained is powerful enough to also solve the specific cases earlier. This might be intentional. The important part is to be able to apply what is taught, and if students always bring out the big guns, they will also always get answers, so who cares if the conflation is happening.
Note that if the contour deformation is to reduce the complicated integral into the stationary phase approximation case, i.e. making the magnitude = constant rather than phase = constant, then it is definitely not a steepest descent anything. But, apparently, the constant phase method is a lot easier to work with and get correct answers, than the constant magnitude method. That is, the steepest descent down a few saddle points is a lot easier to work with than getting all the stationary phase contributions along the constant magnitude contour.
