How did Hedwig know the Dursleys would have a Christmas present for Harry?

Question

We know that Harry receives a present for Christmas from the Dursleys even while he is at Hogwarts. Here are the quotes I remember:

A second, very small parcel contained a note.
We received your message and enclose your Christmas present. From Uncle Vernon and Aunt Petunia. Taped to the note was a fifty-pence piece.
_{Harry Potter and the Philosopher's Stone Chapter 12: The Mirror of Erised}

She nibbled his ear in an affectionate sort of way, which was a far better present than the one that she had brought him, which turned out to be from the Dursleys. They had sent Harry a toothpick and a note telling him to find out whether he’d be able to stay at Hogwarts for the summer vacation, too.
_{Harry Potter and the Chamber of Secrets, Chapter 12: The Polyjuice Potion}

Harry’s other presents were much more satisfactory than Dobby’s odd socks — with the obvious exception of the Dursleys’, which consisted of a single tissue, an all-time low...
_{Harry Potter and the Goblet of Fire Chapter 23: The Yule Ball}

We see that in Philosopher's Stone, their present is part of a reply to a message Harry sent them, presumably to tell them he'd be staying at Hogwarts for the Christmas holidays.

We also see that in Chamber of Secrets, Hedwig brought Harry their present. But how did Hedwig know the Dursleys would want to send Harry a present? The Dursleys had no means of summoning Hedwig. If she was smart enough to figure out it was Christmas and Harry should receive a present from them, she might be smart enough to figure out the Dursleys hated Harry and they wouldn't send Harry a present (I don't think toothpicks, tissues,etc. can be termed as presents).

In Goblet of Fire, there is no mention of how the Dursleys sent their present. Its obvious the Dursleys would rather eat their television than search for a wizarding post office just to owl Harry his present. Assuming it was Hedwig again, how did she know and why would she make the trip?

My question is:

How did Hedwig know the Dursleys would give Harry a present?

Answer 1

Don't forget that the only person we know is getting notes sent home is Harry, who for our purposes is essentially a muggle born, in that his "parents" are muggles and have no access to an owl.

How would muggle-born students receive their Christmas presents, unless Hogwarts/the student themselves sent their personal owl, or the school owl to their parents, so their parents could then use said owl to reply?

Since your quotes clearly says Hedwig was sent WITH a note,

A second, very small parcel contained a note. We received your message and enclose your Christmas present. From Uncle Vernon and Aunt Petunia. Taped to the note was a fifty-pence piece.

it seems that the school automatically sent out owls to all the muggle born students' families who would otherwise have no access to an owl for Christmas.

Answer 2

So this could be more of a guess, but based on how well the wizarding community is hidden within the muggle one, I think it's very likely that if muggles mail something in their ordinary mail there will be a wizarding person at the post office to separate what should then be posted by owl. This ensures that nothing looks out of the ordinary to onlooking muggles.

the durleys would mail the gift through their muggle system marked for hogwarts. being intercepted at the post office. then being mailed to hogwarts by owl. then the hogwarts owls or personal owl deliver to the individual students unless its a send and receive immediately situation, as Sirius and harry often communicated.