The idea of Clan warships fighting limited trials comes from a small bit of text in FM: WC, in the Clan Snow Raven section. It basically says that most Trials involving warships are limited to some criteria short of complete destruction.
It's clear that that in this reference that it's referring to fairly routine and small scale Trials of Possession. E.g. I want to fight you for that dropship and its cargo. I want to fight you for the rights to mine this asteroid.
Keep in mind the Ravens had greater incentive over the generations to use their warships in combat than most other Clans. They had a lot of them, they had some big shipyards for repair/construction, they specifically bred warriors of the Naval Bloodlines for that purpose and they trained their warship crews to a particularly elite pedigree. You can't expect a Sukhanov bloodnamed warrior to go their entire warrior career having never fought in a live fire Trial from their warship.
That's very different from the other Clans which used aerospace pilots. So the warriors got to get some battles in from a cockpit before making the transformation into warship officer. At that point, especially if they already had a bloodname, they had a solid codex and could shift their emphasis from achieving as many battlefield victories as possible to something more strategic like command of one or more warships, which is a big responsibility.
I think it's entirely possible that the Ravens were one of the few Clans routinely going around issuing Trials of Possession via a warship on something like a regular basis.
So the Ravens had this down to a science, on how to use their warship fleet in combat without decimating it. The other Clans maybe not to their extent but forced to play the Ravens game (and sometimes borrowing pieces of the Ravens' playbook). They might have played it a little faster, a little looser, more easily skewing toward either bidding away the warships from the upcoming Trial, or throwing them into a very destructive battle with no limitations. If the two Clans were in "feuding" status this became more likely, such as between the Blood Spirits and the Star Adders.
FM: WC, page 31 gives us another good reference to a limited Trial in the Clan Cloud Cobra section. It tells us of a 3-warship Smoke Jaguar fleet showing up in Cloud Cobra territory trialing for repairs from a Cloud Cobra recharge station, the Cobras sent in a single Liberator-class warship and some assault dropships. Ten minutes later the Jaguars accepted hegira, left behind a Fredasa warship and Miraborg dropship as isorla and left. It's a very brief reference but its an excellent case in the kind of limited Trials warships sometimes fight.
Engagements against Spheroids don't follow Clan Trial procedures so that didn't apply. Even then we did see a Whirlwind surrender to Taskforce Serpent, the Clan warrior commanding the warship declared that his ship was damaged and unable to continue to fight effectively (sounds a lot like a Clan style Trial in mentality actually...)
As Colt said, the Ravens and Vipers got real nasty and bitter and tended toward a more destructive mentality.
At Alshain, the Ghost Bears overreacted. The Nova Cats showed up for an orderly Trial but the Ghost Bears refused the Trial, took it as something much more Spheroid-like (aka an invasion) and just went to attack speed and opened fire. The Ghost Bear commander later did the Surkai thing (rite of forgiveness). So we just don't know exactly how the Nova Cats might have tried to conduct the Trial, with what specific ships, what conditions etc. The normal bidding did not take place. According to the Nova Cat commanding officer on-site the other warships were just coming to watch the Trial. That being said a fight to the death of some number of warships (whether 1-v-1 or a few more than that) is still probable here because the prize (a Leviathan) was so big and valuable, it was not an insignificant objective to fight over.
When we get into the Wars of Reaving era, limited Trials go out the window and Clan warships start blowing up left and right. Keep in mind with so many Trials of Abjurement and Annihilation going on, that changes the stakes from a limited Trial of Possession to a true life or death situation for the warrior, unit or Clan. Plus when it came to eliminating the "tainted" Clans nothing like Zell was applied, it was scorched earth tactics, even orbital bombardment was used frequently.
The Wars of Reaving were all the way on one extreme but we do have at least one meaningful precedent for that kind of full-fleet mobilization. When Clan Smoke Jaguar went to absorb Clan Mongoose, it resulted in at least one major naval battle above Circe involving like 30 warships in a death struggle. So when it's not a small objective, when its a major objective, a big deal and particularly life-or-death for the Clan as a whole, all the warship nastiness seems to come out. Which makes sense, if you are on the receiving end of a Trial of Abjurement or Absorption or Annihilation, you are going all-in on the upcoming fight.
