Update: Spoke to customer service at American Express again this morning. At first, after a lengthy wait, they said that if my first use was during the June 1-30 period, I would get the benefits of the 10,000 point offer, even though I had applied before June 1. I asked them if they were 100% sure. They then hesitated and connected me to a "special agent", who then told me to go ahead and apply for a new card online; he said I could keep or cancel the previous card, that was up to me. I asked whether there was any limit to number of cards applied for under this promotion, and he said no, there was not.

I am still a little skeptical by the kinds of answers I am getting...