I would appreciate you to confirm or possibly correct:

- For each skill and character there is some sort of timer, which ensures that once you've skilled up you cannot skill up on the very next occasion. This timer, however, can be circumvented by saving and reloading.

- Every skill-up is like a dice throw, i.e. each time you cast Light there is some statistical chance you skill up; this probability gets higher the higher your effort is (higher spell power level, for Artifacts a higher level item is identified OR used, or for Mythology a higher level monster is identified, etc.) and (I think) the lower the skill is. So the higher the skill already the less likely you are to increase it further. From my understanding, however, there is no time dependency (apart from the timer mentioned above and some other recast timers like those Portal spells), so it doesn't matter if it's the first time you cast a spell without skill increase or the 10th time.

- For identifying items with Artifacts every time the whole party gets a dice throw, even though only a single character is attempting the ID. However, only those characters of the party get the chance to skill-up who already have the high enough skill to be able to identify the item at all. So, if only one character of your party is able to ID Renewal Potions, only he will get the dice throw by examining the item (provided the timer for his skill is reset), but if you pick up a Mod. Heal Potion, probably every character will the the dice throw (however, with a lower probability of success due to the lower level item). I even think (not sure though) for low level items, skill increase stops completely if your skill has reached a certain level. This whole thing probably works similiar with Mythology (and to some extend also Communication).

The last point could get you into trouble if there's a big gap of Artifacts skills within your party (for example if you have a Bishop and pick up some RPC with very low Artifacts at some point). Then, if the game at some point produces only a limited number of lower level items, this one guy will have a hard time catching up and get his Artifacts skill to a reasonable level, at least by means of identifying itmes (he can still use magical items).

My understanding is a bit different from yours (assuming Wizardry 8; remember to specify which game you're talking about):

Only certain spells have practice timers. After casting the spell, you must wait a certain amount of time (or save/load) before casting it again. Other spells do not, though healing only counts if the character actually needs healing, and many spells (like Bless, but not Summon Elemental) only count if cast in real combat (in other words, entering combat mode when there are no enemies around just to cast the spell doesn't work).

For skill-ups, I believe the game uses a combination of skill experience and a dice roll. When practicing the skill, you get some skill experience, and then the skill has a chance of increasing based on the amount of skill experience you get. In other words, the longer you go without the skill actually increasing, the better the chance of an increase.

Of note, if you have a character with 15 Alchemy, you can get unidentified Heavy Heal potions easily and right click them to practice Artifacts. Alternatively, give the character an Amulet of Static and use it during combat for practice; the power of a spell doesn't seem to affect Artifacts experience gains. (It does matter for spellbook/realm skills, but not power cast.)

For most skills, increase appears not to be limited by a timer,
(unless the delay is only fragments of a second and cant be noticed;)
e.g. there is no limit for all physical combat related skills, you can pump them from 0 to 100 in a single batlle.

Casting spells at higher PL may inrease the chance, but certainly not multiply it according to increased PL,
i. e. its not doubled if you spam divine trap spells at PL2 on a chest instead of PL1.

I suspect that it multiplies the skill experience you get, but doesn't give you an extra check, leading to the results you observe.

For example (using artificial numbers):
Cast spell at PL1: 10% chance
Cast spell at PL1 twice: 10% chance, then (if failed) 20% chance. Total chance of at least one increase: 28%. (There's a tiny chance of 2 increases here, but it is less than 1% unless the skill is so low (like 0) that it's harder to raise.)
Cast spell at PL2: 20% chance.

Of note, if my theory is correct, it should be possible to modify a save file in such a way that a skill is guaranteed to increase on its next use.

It would be really interesting to know whether there's a timer on, say, Locks & Traps as well. It sounds logical that there is none (since I've never experience a benefit of reloading in this case), but I'm still not sure.

I don't think so. Sir-Tech games always featured repetitive high frequency skill usage grinding. However: spamming Enchanted Blade is as useless as healing full health characters. There's no effect. That's because the spell effect has a timer I reckon. Whether or not this can be circumvented I do not know yet I dare say it would be an inefficient way to grind anyway. Although I do not know details my money is on a non-random skill increase. Try to count the spell usages between levelling up, divided by success or failure (so use something that always succeeds) if you care enough. Maybe higher SP investment helps, had that impression at times. In any case spamming low level spells works better. Maybe it is a base value per success plus a SP dependent modifier? Amulet of Static helps a lot.

In general the most decisive things are the skill usage's frequency, success and the primary attribute required for the skill.

Sure. As for the Amulet, I agree that it's a great early game item, but you can use it like 9 times max and then you have to find a Vendor to sell/rebuy. 9 times is plenty for an Artifact, but it's not much for training.

Revisited this and wondered: how do you train Artifacts anyway? Specific routines you follow?
I routinely divide stacks of unidentified items, possibly storing them in a chest, which does help but won't skyrocket your Artifacts Skill. Amulet of Static is pretty much my best active training, simply depleting it in about every fight in Arnika while also using two of those on the road. It isn't powertraining of course, more of a mini game along the way. With the Amulet of Static being really powerful early on it has multiple benefits and is easily done. This early routine still does a lot over time.
My Mage actually had problems identifying high level items despite identifying every found item herself. Somehow I hadn't thought of it earlier and didn't bother using the Amulets too much... I can't remember having had that problem previously. Then again there are few things you really need to identify. I probably never cared about boosting Artifacts to actually high levels. A high level Mage Spellbook needs to be identified to be used though. Amulet of Static was a decent first aid but I didn't find any real powertraining and actually invested some level up points there.