rob does a good job of covering the overall use of the tools. I find these discussions interesting since it is possible to be right but have different answers. All of these types of saws function in the same way. Some are better at dealing with intricate work and some are designed for rough jobs. It does not help that the names of these tools could be interchangeable (reciprocating saw being the worst offender of vague tool name).
I have always understood the first tool you show as a scroll saw. It is known to have a blade fixed on both ends and is, typically, used for intricate work. The unit is fixed (not hand held or particularity mobile) and you move your work piece around the table. Similar to a band-saw, the blade and work piece are in cleaner view than a hand held jigsaw. I have always understood these to cut at 90 degrees given how the tool is constructed. This is by design since the arm does not rotate. Some have tilting tables to make up for this.
I consider the jigsaw to be the middle ground between a reciprocating saw and a scroll saw. The jigsaw is represented by the other pictures you have shown. It's a smaller reciprocating saw and not fixed like a scroll saw. You can use a variety of blades of varying sizes and material designates. The base on many modern jigsaws pivot to allow angles to be cut. The mobility is arguably its greatest asset.
As I tried to cover the grey area between sabre saws and jigsaws, I'm with rob in that I do not think reciprocating saws fall into this category. In short sabre saw was a common term to describe a jigsaw and appears to have evolved in name. Older woodworkers still refer to jigsaws as sabre saws.
Reciprocating saws are meant for rough work to get the job done quickly. I have used them in demolition for studs and nails. I own one today for help with pallets. They don't generally have a place in fine woodworking.
The real reason I wanted to post this is I found more old ads that might explain the evolution of tools and terms. The first I found interesting since it is for a scroll saw from 1880's. Besides being foot pedaled it looks much the same as its powered modern descendant. I have not found any conflicting information as to this tool's name over time.
Image From AtticPaper
More on the sabre saw/jigsaw mix up
Images from Kijiji
I found this on sale at Kijiji. The great thing about it is that it is labelled as both a jigsaw and sabre saw. I could see this as an attempt to remove the potential confusion between tools' names but that is just speculation. I still think the fight is over and both side are right for this one!
One of the earlier models I was able to locate of a "Recipro Saw". Specifically the "Skil Model 577 Recipro Saw". Very similar design to the modern reciprocating saws we see now. Hand placement is relatively the same as well.
Image Source: eBay
These tools all function similarly. Each is better suited for certain work and woodworkers can easily have a bias towards one over the other. As I have seen here at WW countless times is that people will recognize the same tool under a different name but they are not necessarily wrong.
While I didn't really cover it, the blades are what can make the biggest difference when it comes to how tight the angles are that you can make, how clean the cut is and in some cases depth of cut.