Why didn't the Ministry detect Dobby Apparating?

Question

In Chapter Two of Chamber of Secrets Dobby smashes the pudding and Disapparates:

The pudding fell to the floor with a heart-stopping crash. Cream splattered the windows and walls as the dish shattered. With a crack like a whip, Dobby vanished.

Harry immediately receives a warning from the Ministry for his alleged use of underage magic, but it only mentions the Hover Charm:

We have received intelligence that a Hover Charm was used at your place of residence this evening at twelve minutes past nine.

Why didn't the Ministry also detect the Apparating?

(This surely would have been relevant, because while a Hover Charm in Harry's house can easily be attributed to him, Apparating would seemingly be indicative of a more advanced wizard's involvement.)

Answer 1

Elf apparition is unique to elves and is not the same as a wizard's. Hence it may not be detectable by the Ministry, the same way Wizard Enchantments cannot prevent them from Apparating or Disapparating.

House-elf apparation is unique to house-elves; it is a different form of magic:

“Elf magic isn’t like wizard’s magic, is it?” said Ron. “I mean, they can Apparate and Disapparate in and out of Hogwarts when we can’t.”

There was silence as Harry digested this. How could Voldemort have made such a mistake? But even as he thought this, Hermione spoke, and her voice was icy.

“Of course, Voldemort would have considered the ways of house-elves far beneath his notice, just like all the purebloods who treat them like animals. . . . It would never have occurred to him that they might have magic that he didn’t.”

Ron says that House-Elf's magic is unique to elves, at least in regards to apparation. Although Ron may not know much about house-elves, Hermione confirms that "Elves have magic that wizards do not."

Another proof that the magic is different: Whenever house-elves apparate in the books, they never need to spin on their heel.

What is the source of this unique magic? it seems inherent to the magical creature:

“The house-elf’s highest law is his Master’s bidding,” intoned Kreacher. “Kreacher was told to come home, so Kreacher came home. . . .”

Ron seems to have understood as well that by House-elves it's instinctual:

“Wish I could Disapparate like a house-elf,” said Ron, staring at the spot where Dobby had vanished. “I’d have that Apparition Test in the bag.”

(Of course, this could be interpreted differently.)

It seems that Wizards cannot stop a house-elf from Apparating or Disapparating. Even the Malfoys do not have a charm stopping house-elves from apparating into their Manor, and presumably, unlike Voldemort, they would've considered the possibility, since they themselves owned Dobby in the past.

Dumbledore is in charge of the Hogwarts Apparation charm:

“As you may know, it is usually impossible to Apparate or Disapparate within Hogwarts. The Headmaster has lifted this enchantment, purely within the Great Hall, for one hour, so as to enable you to practice.

Yet he too does not enchant Hogwarts to disallow elf apparation.

From all these cases, and from the fact the Ministry did not detect Dobby's apparation, it seems apparent to me that Wizards are in fact unable to prevent or detect House-Elf Apparation.

As for the Hover Charm, perhaps it was a wizard charm that Dobby learned from the Malfoys, and thus detectable.

As an aside: Interestingly, Elves aren't the only animals who can apparate without human restriction. Phoenixes can too:

Fawkes circled the office and swooped low over him. Dumbledore released Harry, raised his hand, and grasped the phoenix’s long golden tail. There was a flash of fire and the pair of them had gone. “Where is he?” yelled Fudge, pushing himself up from the ground. “Where is he?” “I don’t know!” shouted Kingsley, also leaping to his feet. “Well, he can’t have Disapparated!” cried Umbridge. “You can’t inside this school —”

Answer 2

The Trace is only set up to detect wizard's magic, not house-elf magic.

That leaves the question of why it detected the Hover Charm, but we know that house-elves are capable of casting wizard's spells as well as using their own special house-elf magic, because Winky was accused of using Harry's wand to cast the Dark Mark in The Goblet of Fire.

It wasn't true, but the fact that the accusation was considered at all seriously indicates that house-elves are known to be able to cast wizard spells. For a powerful spell such as the Dark Mark, they would need to have stolen a wand, but even untrained wizards can perform simple magics without one.

We can conclude that when Dobby levitated the cake, it wasn't house-elf magic at all, but just a perfectly ordinary Hover Charm, subject to the Trace regardless of who performs it. That's a very simple spell, basic enough to be taught to first years, so an inherently magical creature would certainly be able to cast it without the assistance of a wand.