Harry didn't do magic, why does the ministry think he did?

Question

In Harry Potter and the Chamber of Secrets when Dobby uses the Hover charm, Harry is warned that he isn't supposed to do magic outside school.

However, it was Dobby who did the magic.

Don't they know who exactly does magic? What if some grown-up wizard came to Hermione's family and did magic in the presence of the muggles, would they immediately suspect Hermione and send her an owl?

Besides that, Harry could have gone to court against the ministry and presented his wand so they would check to know that his last spell wasn't the Hover charm. As far as I am aware it is possibility and he wouldn't be guilty.

Yet, in the 2nd or 3rd movie Harry does the Lumos charm under his blanket, isn't that disallowed too?

So, my thought is that the ministry has problems with spotting magic and the rest.

I'm asking why they thought Harry did it and why they sent him a warning when should first have done an inquiry and therefore would know that it indeed wasn't Harry who did the spell.

Answer 1

In the books, things are made a lot clearer (Even if not fully clear).

The ministry apparently can NOT identify who actually did the magic, just that magic happened around an underage wizard. In case of muggleborns, there are no possible other sources around. In case of wizardborns, the ministry trusts parents to make sure there is no underage-magic going on.

"Dobby,” growled Harry; this injustice still rankled. “So if you’re underage and you do magic inside an adult witch or wizard’s house, the Ministry won’t know?”
“They will certainly be unable to tell who performed the magic,” said Dumbledore, smiling slightly at the look of great indignation on Harry’s face. “They rely on witch and wizard parents to enforce their offspring’s obedience while within their walls.”
_{(Harry Potter and the Half-Blood Prince).}

Harry reading under the "Lumos"-light was a movie-only addition! The movies are notorious for adding inconsistencies like that. The nature of the homework they get over the holidays seems to be more along the lines of "write an essay" and "read that book", so no magic would be necessary.

Answer 2

Elf magic is quite fickle.

House-elves have got powerful magic of their own, but they can’t usually use it without their master’s permission.

House elves can apparate in and out of places where wizards can't. Anti-apparition wards don't mean anything to them. The Trace is able to detect any magic done near an underage wizard. Elf magic can be detected, as shown in the second book, but it is not identified as such. The Trace assumes that the magic was done by the Wizard. However, when magic is performed by an adult wizard around an underage wizard, it will be attributed to the adult wizard.

“Incendio!” said Mr. Weasley, pointing his wand at the hole in the wall behind him.

Here, you can see that Arthur Weasley performs magic in front of Harry without triggering a underage-magic warning. This is also why underage magic can be performed in wizarding households with wizard-parents.

Therefore, Elf magic can be detected, but cannot be identified as such.