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Interesting question about the different tourney rules as seen in "Hedge Knight" and the novels.
It was not so much a question of some king changing the rules, as you venture, as it was of the rules themselves being very variable. Medieval tourneys were never governed by a single set of rules or rulesmakers, like NCAA football or major league baseball or even (shudder) boxing. In essence, every tourney had its own rules. The lord or king who was staging the event would usually choose the format of the tournament in the broadest sense, and then appoint a "master of the games" to run the event and make all the "fine print" decisions.
The earliest tournaments were melees for the most part, fought over miles of woodland by teams of knights. It was a battle, in essence, though with blunted weapons (usually). The lists came much later, and the formalized joust as we know it through IVANHOE and other popular sources. But even then, there was no standardization. Sometimes they still fought as teams; fifty Scots riding across a field at fifty English. Sometimes they had very elaborate pagaents, like one "Tournament of the Golden Tree" in which a team of champions had to defend said tree against all comers, each trying to snatch a leaf (the gold leafs were gold leaf, so to speak).
In the real world, heralds eventually assumed a great deal of responsibility for tournaments as time went on, and by the 15th and 16th century things were getting rather formalized... not just in tourneys, but in regards to all the forms of knighthood and heraldry. Before that, however, you had much more diversity.
In the case of the Seven Kingdoms, basically I am going with the earlier more diverse model. Over the six books (and whatever additional Dunk & Egg stories I write), I mean to show a nice range of tournaments.
Of the ones so far... well, the Hand's tourney at King's Landing was put together hastily, on Robert's whim, and so was relatively small, which allowed the single-elimination tilting format, which your opponents are chosen simply by the luck of the draw, and only one champion remains at the end. I also used the free-for-all last-man-standing style of melee, which did not exist in the real world so far as I know (melees were mock battles fought by teams), but which I thought offered juicy possibilities for a fantasy book.
At Ashford, instead of the single elimination, I went with your basic IVANHOE champions-against-challengers format, which suited the story better (I wanted Dunk to have to risk all he had going up against one of five champions, rather than simply drawing an opponent by lot, and maybe catching a break. If the champions are as formidable as the Fair Maid's were at Ashford, the challengers face a much more difficult task than if they stand to draw any foe in the field). And I confess, I have always loved the scene in IVANHOE where the Disinherited Knight rides down the line and knocks down all five Norman shields, and wanted to do my version thereof.
Renly's little tourney was pure free-for-all melee again, because it was a very impromptu affair. The all-against-all format, with its quick alliances and betrayals, allowed me to show both some of the popular feeling against Brienne and her own quite formidable skills.
Joffrey's little tourney was really more an exhibition of jousting.
. . . the great tournament at Harrenhal during the year of the false spring, the tourney where Rhaegar crowned Lyanna Stark as queen of love and beauty. That was a much bigger tourney than either Ashford or the Hand's tourney. The IVANHOE format again, champions and challengers, but longer, with more challengers... and with a seven-sided team melee in the ancient style. (A lot happened there at Harrenhal. If I ever wrote the prequel book some readers keep asking for, I could probably set the whole thing during those ten days.)
As to your questions regarding the participation or non-participation of sellswords, squires, freeriders and the like, again, I don't see that as the difference as being chronological so much as geographic. The Reach is the heart of the chivalric tradition in the Seven Kingdoms, the place where knighthood is most universally esteemed, and therefore the place where the master of the games is most likely to devise and apply stringent rules. In Dorne and Storm's End and the riverlands and the Vale, things are perhaps a little less strict, and north of the Neck where the old gods still reign and knights are rare, they make up their own rules as they go along.
This has real world parallels as well. In the high middle ages, France was the apex of chivalry. German, English, Italian, and Spanish knights followed the fashions the French chevaliers set, although they did not always get them right. And if you went further afield still, to places like Scotland, Hungary, and Georgia, customs diverged even more.
The personalities of the sponsoring lords and their master-at-arms are another factor. Robert Baratheon was not a great respector of old traditions, and he would hardly have wanted a "knight's only" tournament to honor Ned, who was not a knight. Lord Ashford of Ashford, on the other hand, was trying to curry favor with Baelor Breakspear, the preeminent tourney knight of his time.
As to trial by combat... yes, the Trial by Seven was very much a special case. It was originally an Andal religious ceremony, and even at the time of the "The Hedge Knight" they hadn't fought one for a hundred years (or whatever I said).
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