I'd pick right after June 17th, which is the day you mentioned.
Two reasons: schools are just then ending for the summer, and most people, I think, don't go on vacation immediately after school ends, so June 18th is good.
The other reason is that I've heard that Auguts has the worst weather of the year for Florida. More rain, more heat, more humidity.
I've only ever gone to WDW the last week of June/first week of July, and it was always great.