Choosing an approach for a general purpose CCFL power supply is difficult. A variety of disparate considerations make determining the "best" approach a thoughtful exercise. Above all, the architecture must be extraordinarily flexible. The sheer number and diversity of applications demands this. The considerations take many degrees of freedom.I think this introduction is partly an inoculation to the "summary" tables that are included later in the app note. See footnote 18 on page 49.
Figures 35 through 38 are quite similar to App Note 55 Figures 6 through 9 (although a newer, higher-frequency switching regulator is used in the last two figures).
Figure 39 is a brand new circuit, using a dedicated CCFL integrated circuit, the LT1183. This single-chip solution is the major advancement in this "fourth generation" treatment of CCFL power supplies, and it includes many of the features that Jim has discussed in previous app notes, such as open-lamp protection and the variable LCD-contrast supply voltage.
Note the name of pin 13 on the LT1183, "Royer". Although he doesn't mention it here, Jim commented in a later publication  that "Local historians can't be certain, but this may be the only IC pin ever named after a person."
That's a very good question. Dear loyal readers, is this the only IC pin named after a person?
 Jim Williams, "SMBus controlled CCFL power supply," Linear Technology Magazine, vol. 9, no. 3, p. 35, September 1999.
Tutorial footnote (for new readers): The above power-converter topology is a resonant Royer converter. The LT1183 is a controller chip that drives a Royer converter for this specific application: driving the cold-cathode fluorescent lamp (CCFL) for the backlight in a laptop screen.
The so-called ROYER pin connects to the center-tapped primary of the transformer (which is the unregulated-power input in this topology). In the circuit shown, the pin isn't doing much. However, if the BAT pin and the ROYER pin aren't connected together, then all of the input current flows in the BAT pin and out the ROYER pin, and the chip can sense the primary-side input current in the transformer. Thus, the LT1183 can drive the lamp in a floating (isolated) topology, without any direct feedback to the chip from the high-voltage side.
Here's the explanation from the LT1183 datasheet:
ROYER (Pin 13): This pin connects to the center-tapped primary of the Royer converter and is used with the BAT pin in a floating lamp configuration where lamp current is controlled by sensing Royer primary side converter current. This pin is the inverting terminal of a high-side current sense amplifier. The typical quiescent current is 50μA into the pin. If the CCFL regulator is not used in a floating lamp configuration, tie the Royer and BAT pins together. This pin is only available on the LT1182/LT1183/LT1184F.For more details, see Linear Technology App Note 65: "A fourth generation of LCD backlight technology: Component and measurement improvements refine performance." Appendix K is titled "Who was Royer and what did he design?"
Update: A commenter on another website suggested that an argument could be made for the ZENER pin on the CD4046 Phase-Locked Loop chip. That's a good one. Any others?
Update 2: Further discussion on this topic has been moved to a dedicated post: IC pins named after persons.