Can I drain a 15'x35' concrete brick patio using a floor drain connected to a 4" PVC pipe?
edit:
After reading your updated question, you will be fine installing the smaller drain. I have a large driveway drain that I installed a few months ago, and it works great to drain the huge puddle that used to form. I used a drain box similar to the 9x9 that you have, and 4" pipes.
Occasionally, during a very heavy rain, it will get behind and not completely drain until the rain stops, but that is because the pipe itself is only 4"!

Great question!
I am not an expert, but I would highly recommend the larger drain. The reason is, on a small drain, every time the drain gets stepped on (or driven over with a lawn mower), 100% of that weight gets put on your pipe underneath. A 4" pipe isn't designed to carry that weight, and could crack.

Even though the 9x9 drain won't necessarily drain any faster, it will have more surface area under it to support being stepped on, that way the soil under the drain, instead of the pipe supports the weight.