[Aga-chan]

The inspiration for "July 1st" was not only the birthday party or her day off, but also a photography shown to her during the "Hamasaki Republic" interview.
From masa-san's site:

This article (add: the quote posted by shavoingin) makes it clear why the second song is titled strangely as "July 1st." But the scene sung about in this song is not the party place at night but the sea in the daytime. Why? -- In fact, there was one more thing which lead her to make this song. In the interview of "Hamasaki Republic", she and the interviewer were talking as below, looking at a photo of the sea in a magazine.
　O: How do you think about the sentence? ("If it's fine tomorrow, let's go to the sea.")
　A: Nice. I'm glad to hear someone tell me so, for example. A family member or a friend or a boyfriend. Because if someone tell me so, I'm in his or her future. I'm in his or her tomorrow. I'm glad of that.　　("Hamasaki Republic")

We have to consider that the lyrics who accompaign the picture with "0630" are those for "Greatful days". I don't think ayu would've used this picture if her father had died that day with the lyrics of "Greatful days". It'd be more natural to use it with "Memorial address" then. So I think that "Greatful days" is very much tied to "July 1st" and maybe that's the reason for the "0630" writing. Maybe she wanted to emphasize it.

Sorry, if I didn't make much sense, but I can't think of any other way to put it, now...