The first one is a trombone matched antenna, the second one looks like a trombone matched antenna, but in fact is not. What appears to be a trombone style matching system is actually a horizontal linear loading section on a vertical antenna. It is one of three models I made, another model looks like this...

That antenna predated my efforts to make a tapped trombone matching section that I was/am talking about in this thread. I was experimenting with linear loading with 5/8 wavelength antennas, essentially using that as a loading method to get an electrical 3/4 wavelength antenna, which should combine with 1/4 wavelength radials to produce a low R at resonance, therefore having a naturally low SWR. I did not get it to a perfect match, but using nothing but a linear loading section I got SWR down to 1.2:1. The trombone matching system does, however, outperform that model, if only slightly, it is more efficient according to 4Nec2.
In that case, linear loading essentially has a similar effect as using a loading coil for loading, at least so far as it makes a shorter antenna act like a longer antenna. It was an experiment, and it actually worked better than I expected it to. I modified that model to make the tapped trombone model as shown in your image above it, so if anything my experimenting with a trombone style matching system in a 5/8 wavelength antenna model is an evolution of my playing with linear loading in a 5/8 wavelength antenna model.
The trombone match I have has a tapped element between the bottom and center of the trombone sections, I can also adjust either side of the trombone independently using the variables I talked about above. I have a variable for each side's length, and it is more a matter of changing them a little at a time until I get the match that I am looking for. As I recall offhand, one side has more of an effect on where the SWR minimum was, whole the other side had more of an effect on how deep the low SWR point was.
The DB