If you have a current, active real estate license any advice you've been providing has been 'official' whether you mean it to or not.
Typically rental income to a new owner begins on the actual day he becomes the new owner.
In the case you cite with escrow closing on June 14th the new owner would be be due a pro rated share of the June rental income from June 14th til the end of the month.
All income for July should go to the new owner.
If that is the wishes of both the buyer and seller they can draft an agreement stating all incomes go to the new owner on July 1st but absent any such agreement rents would go to the new owner on that date anyway.
You may be confusing the new owner's payment due date with the rent.
Typically mortgage interest is paid 1 month in arrears (behind) so in your scenario the new owner's first payment would typically become due on August 1st. But because there would be interest due from the June 14th closing date and no payment would be made until August the lender would typically collect the June interest at closing.