I think AmEx could distinguish if they wanted to, but I believe it is the latter part of the previous post where people are running massive amounts through personal cards for this like Google Adwords, resale merchandise, shipping costs, etc. I think it is just setting AmEx with permission to close accounts if needed although I presume they were already doing so even without the added statement.

I do think usual T&E for company reimbursement on a personal card is not going to be an issue, but no one really knows if AmEx changes its mind.