@TuxKey:
Wow, you do seem to have yourself tied in knots, the number of links made my head spin a bit. I agree with Rooster by the way - it is far better to ask questions on the forum thread rather than PMs to several people. Then other people can benefit from the information too.
First, let's understand fundamentals:
The reason that the TS100 doesn't like floating PSUs (floating in this context meaning 2 pin mains or fake 3 pin) is that the output floats to half mains voltage due to the EMC filter caps. It is not surprising that the temperature sensing gets disturbed when the element and thermocouple in the tip are floating at ~115V AC and the surrounding tip is grounded. The thermocouple is arranged to have best possible thermal coupling so it is inevitable that there is some capacitive and resistive leakage current. The thermocouple sensing circuit is looking for mV levels. I suspect that most other cartridge type irons would behave the same, it's just that their PSUs don't put them in that situation. It's as simple as that.
Let's now examine your options:
Firstly two of your links are for the Meanwell LRS100 series SMPS. This is designed to be built into equipment, not used standalone. It has exposed mains terminals and chassis openings that are too large to meet CE if not enclosed. At a minimum, you would need to mount it in an enclosure, either sealed, or with <3mm ventilation holes. You would also need to include a mains fuse on the live at the inlet, not to protect the PSU (which has its own internal fuse) but to protect against fire if a fault develops in your internal mains wiring. I am really not comfortable with you doing this, given your level of experience.
Down to the other two (enclosed brick) PSUs. If you want to adopt the 2 wire silicone cable, with earthing tab as Rooster has used, then you must have a PSU with grounded negative (how else are you going to get your earth reference for the tip with 2 cores?). The only alternatives I can think of is find a suitably thin 3 core cable and separately earth the 3rd core, or use a separate thin silicone tip grounding lead, maybe clipped to the cable at intervals for convenience - I think that was the original idea of the TS100 designers (use a grounding lead when using for sensitive electronics, don't bother with a battery supply or non-sensitive stuff).
Have measured the voltage on the barrel of the adapter you have bought relative to mains earth? If properly designed, then the EMC filter leakage currents will be returned to the input mains ground (3rd) pin, as shown in the block diagram you attached. If so, the output will be floating near ground and will be suitable for you to ground the tip using a separate grounding wire.
If you choose to return it and get the grounded output version instead, then you can adopt the 2 wire cord and grounding arrangement as shown in Rooster's photos.
Your final alternative - and the one that I suggested to Rooster (and you too, I think) and the one that he adopted, is to purchase a surplus genuine 3 pin brand name laptop PSU. As I have previously said, Genuine Lenovo 20V 3 pin units can be purchased on ebay very cheaply. Similar offerings from Dell, HP etc too. Ok you won't get the maximum 24V but with the thermal regulation, you are unlikely to notice the difference. How you attach the flexible silicone cord to the output is entirely down to your preference, either M-F connectors or cut the wire and solder+heatshrink. It makes no difference.
I, myself, have used a laptop PSU. Ok, I have internally modified it to put out the maximum 24V but this is not something that I would want you to do. It involves opening the sealed casing, reverse engineering, and making circuit modifications. If done wrongly or carelessly, it will compromise safety.
I hope the above gives you a full understanding of the reasons, issues involved, and your options. If not, then do please ask.