If you want a fix that's going to last for some time to come, I think the only option you have is to install at least 2 drains, and pipe them out to daylight. Meaning, renting a saw and cutting trenches at least 10" wide to both puddles, and extending that trench all the way to the closest edge of the patio. Install pipe and drains, fill back over pipe with stone or gravel, install a few dowels into the existing concrete, and pour and finish concrete into said trenches.
Cutting extra control joints rarely, if ever, works long term, because their too small and fill with deris quickly, and need constant cleaning.
Installing a dry well, like you mentioned, rarely works either, unless the ground underneath takes water very quickly.
I should mention as well, THIS is the exact reason why ALL exterior concrete needs adequate pitch. I've been asked numerous times over the years to do half court + basketball courts in customer's back yards, and I hardly ever get the job because I insist on 2% pitch in one direction on all exterior concrete. Most homeowners can't grasp having their kids possibly play on "THAT MUCH PITCH!" Someone else gets the job and they end up playing in water puddles, like this, instead..........
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