I don't think there is a hard and fast answer here (at our current state of knowledge). I tend not to put a lot of trust in websites, unless there is corroborative evidence from reputable sources (or the website is itself run by people who are both reputable and can provide sustantiated sources for their points). That might seem pedantic but then I come from an Academic background where sources for opinions need to be advanced and open to question.
It's not an area that I have pursued in any particular fashion, but I am aware from original sources that the symbols were in use in the sixteenth century. A little bit of research in my library finds this information from James Herschel Holden's 'A History of Horoscopic Astrology' (1996), page 14 , referring to 1st Century AD sources, Holden says 'In other original sources the Sun is represented by a small circle with ray attached and the moon as a crescent but no symbols are used for the planets.
Then in a footnote on the same page (footnote 16) he says
O. Neugebauer & H. B, Van Hoesen Greek Horoscopes (Philadelphia: The American Philosophical Society 1959) pp 19 and 163 point out that symbols whereby the planets and signs are represented today first appeared in medieval MSS. And it might be added that the mauscipt symbols differed somewhat from their modern forms, which were regularized by 15th Century printers. The mauscript symbols of the signs were pictorial, while the symbols for the planets were simply modifications of the Greek letters of the planetary names, e.g. 'K' for Saturn (Kronos), Z for Jupiter (Zeus)...etc. They were adopted merely for convenience and had none of the esoteric meaning that some modern astrologers have attributed to them.
It's also clear that in the md 16th Century Lilly used the symbols as shorthand abbreviations and attributed no overt esoteric meanings to them.
So Sun and Moon seem to have the longest history. The medieval period saw the development of symbols for planets and signs (probably working on the use of initial letters) and these were standardised by the printing industry and in common use by the sixteenth and seventeenth centuries.
Who specifically designed them? I doubt if anyone can claim that honour - in the sense of designing them from scratch. Instead they probably just developed over time and form was imposed by the needs of printers.