If you can afford being unemployed for 3 days and if you would like to leave your current company with the best impression, I would give notice today.
You mention the job is causing you stress and anxiety, and you already have your next job lined up.
Just looking at the calendar: today is June 20, so if you have a 2-week notice today, your last day of work would be 4 July. You have a confirmed start date of 10 July; assuming you're working a usual M-F schedule, that would leave you between jobs for a total of 3 days, (5-7 July).
Since your boss is leaving for two weeks starting tomorrow, giving notice after he's out of the office might be a way to avoid confronting him. Not sure if that's your goal or relevant to the objective of your PIP. Not having that interaction may also be viewed as cowardly; we can't read your boss's mind or what his issues are with your performance.
I would give notice phrasing it this way: "I anticipate my last working day to be 04 July, but am willing to work through 07 July to facilitate the transition, if requested."
Depending on your company's policies and whatever you've got going on with your PIP, the company may ask you to not come back the day you give notice, so be prepared for that, as well.
All that being said, there's another way I could be reading your question: have you been purposely (consciously or unconsciously) giving your company reasons to terminate you so you can collect a severance? Already having another job lined up, it seems like you've been preparing to leave anyway. Are you playing chicken with your company, seeing who blinks first, hoping to collect a severance on your way out the door?