There are several versions of the triplicities. The very first made no reference to element at all, thus instead of the Fire triplicity, Astrologers referred to the Aries/Leo/Sagittarius trigon. Elements were incorporated in the second century CE, mainly due to the influence of Vettius Valens. In both versions the trigon rulers were the same, with three planets listed for each trigon. To use the elements: Fire and Air made use of the diurnal planes, Sun, Jupiter and Saturn as Triplicity rulers, plus the use of Mercury for Air by night. As you have seen the order is not the same, even though the same four planets are used for both triplicities. In each case the Triplicity rulers have either rulership or exaltation in the triplicity. Sun has domicile rulership in Leo and exaltation in Aries. Jupiter has rulership in Sagittarius, Saturn rules Aquarius and is exalted in Libra; Mercury rules Gemini.
The Earth and Water Triplicities used the three nocturnal planets, Moon, Venus and Mars. Again the order is not the same in both cases. Venus does take the day rulership for both being the ruler of Taurus and exalted in Pisces but Moon rules Earth by night (don't forget she is exalted in Taurus) and Mars takes the Water Triplicity by night ruling Scorpio. Despite Mars and Venus having rulerships in Fire and Air signs, they can't rule the respective triplicity because they are nocturnal planets. The same goes with Jupiter and Saturn. They can't have rulership in Earth or Water triplicities. This system is often referred to as the Dorothean Triplicities after Dorotheus of Sidon, whose book Carmen Astrologicum passed on to the early Medieval Astrologers. However Dorotheus did not invent the system, it was used before his first century work.
This system seems to have been used universally throughout the Hellenistic period, although Ptolemy produced a variant during the second century. Most Medieval Astrologers continued to use this system until it began to find it's way into Western Europe in the twelfth and thirteenth centuries. Ptolemy's work was the only one that remained available throughout the period from second century to the late medieval period and that became the system that Western European Astrologers began using. It was the system that Lilly adopted because he mistakenly believed that the Dorothean Triplicities were an Arab interpolation (Dorotheus was only available as an Arabic translation of a Persian translation of the original Greek and if you read it, you'll understand why it seemed to be primarily an Arab work.)
Ptolemy's triplicities had only two rulers and the main difference is that he reduced the Water triplicity ruler to Mars both by day and by night.
You asked what the role of the third triplicity ruler was. In forecasts for how a person's life would develop the Hellenistic Astrologers had many methods, some of them quite sophisticated (more sophisticated than many used by modern Astrologers). However a simple method was to divide the life into two parts (not necessarily equal). The first part of life was ruled by the first triplicity lord of the Sect Light, and the second part of life was ruled by the second triplicity lord. The participating lord supported both rulers - that is it had a secondary role to play, throughout life. So if you were born at night, like me, you looked to see where the Moon was, and identified it's triplicity. In my case the Moon was in Aquarius (7th House) and its triplicity rulers were; Mercury, Saturn, Jupiter, for a night birth. So Mercury rules the first part of my life, Saturn the second and Jupiter helps out either or both. The condition of these three planets will indicate how successful I am in those two parts.
During the Medieval period, this changed to a life being divided into three parts, with each triplicity ruler taking it's appropriate part. So for me, that would be Mercury first part, Saturn second part, Jupiter last part.
This use for forecasting is not a part of horary.
Lastly there's the importance assigned to a planet being in one of the signs of the triplicity it rules, according to the ruling sect. This is the only use of sect for most horary readers, indeed some modern readers use the day rulers no matter when the chart was cast, or don't bother with them at all. Lilly only gave 'points' to a planet that was in its own triplicity and was the ruler of that triplicity by sect - for example Mercury in Libra in a night chart. Only Mercury secured the points. If Saturn was also in Gemini, Saturn didn't get any dignity points because it was the day ruler for Air but it was a night chart.
It seems that in the earlier Medieval period either both planets scored the points or Saturn was not treated as being Peregrine (no penalty points for being out of dignity). The Hellenistic authors didn't score at all but a planet that was in an alien triplicity was not as strong as a planet in a friendly triplicity. So Saturn being placed in Taurus was not in a friendly triplicity - Earth is ruled by nocturnal planets and Saturn is diurnal. Saturn in Fire or Air is in a friendly triplicity (ruled by diurnal planets). So Saturn is qualitatively stronger.
The Trigons in Hellenistic Astrology were more important than the Triplicities in Lilly's time, as were the Bounds (or Terms) and the Triplicity rulers played a greater role. The Trigons/Triplicities were dignities from the very start of Horoscopic Astrology. They're not an add on. You'll find out a little more information if you use the link to Essential dignities in the notes section.
