The important thing is that pH 4 solution is nearly distilled water and it will leach the KCl out of your probe which will likely shorten its life.
A pH 4 solution can be distilled water with a wee bit of, for example, HCl in it. Or it could be a strong solution of sodium sulfate with a wee bit of sulfuric acid in it. The important point is that if it does not contain potassium ion and chloride ion there will be a chemical potential difference between the gel in the reference cell and the storage solution and those ions will migrate out. Thus you must, for the conventional KCl, Ag/AgCl reference meter, have KCl in the storage solution. But meter technology keeps advancing. Ag/AgCl with a ceramic frit isn't the only reference cell design out there any more. Therefore, you store it according to the manufacturer's recommendation. If he says store it in mild kimchee, store it in mild kimchee, even if you prefer the spicy.
I can't tell if that probe is double-junction or not.
And it doesn't matter. Twenty years ago I would strongly recommend a refillable, double junction, renewable junction electrode absolutely. With modern design none of the above seem necessary. One of the biggest modern improvements seems to be the 'rag junction' as I call it because I can't remember the name of the material but it looks like a piece of rag. This has become quite popular and reduced the cost of good electrodes substantially.
The best electrodes I have ever used are single junction. I just can't kill these things. I finally threw the one I had been using for over 5 years out simply because I had a new one sitting on the shelf assuming I'd never get 5 years out of a pH electrode.
You can read more about what I think about all-in-one pH meters on the Bru'n Water Facebook page.
I'll tell you what I think of them and, as I don't do Facebook, I'll tell you right here. Technological marvels! That is, of course, provided you get a good one. Depending on what you are trying to do there are some potential disadvantages. Most of them are low cost intended for spot checking of things here and there around the plant. As such they don't have features like mV display mode, ability to store data in memory, communicate with a computer and, the main one, a probe narrow enough to cram into the narrow neck of a graduated cylinder or small reagent bottle. Other than that they are fine and represent absolutely as good a source of accurate pH meter readings as comparably priced benchtop meters. See
https://www.homebrewtalk.com/f128/ph-meter-calibration-302256/ for some tips on pH meter calibration or use but in this context for the stability graphs (which are the ultimate reflection of pH meter accuracy).
As I don't do Facebook I can't see what Martin thinks but he used to have some interesting theories about ATC being detrimental in an inexpensive meter because it forced the manufacturer to put money into ATC components that would have been better spent on the electrodes and other electronics. Don't know if he still thinks that way but if he's still does, ignore it.
BTW, this is not intended to convince anyone that he should by a pocket meter but rather to exhort people not to let anyone convince them that they shouldn't. A well made pocket meter (e.g. Hach and quite possibly this Omega) will give you readings just as good as a siimlarly priced benchtop unit. Buy whichever you prefer.