It's still foundation drain water. Some water is weeping into the pipe, some isn't due to the large volume. It's making it's way through the gravel nonetheless.

I'd make sure gutter spouts are extended away from the house Later, maybe add a gradual slope away from the house, too, if your situation permits.

Let's hope you don't ever lose power with that kind of volume.

yes do yourself a favor and get a backup power source for that pump asap. only a matter of time before you lose power during a big storm.

here's the system I have it works awesome and comes with heavy guage hook up wire:
http://dcacpower.com/2000-Watt-12V-DC-to-110V-AC-Royal-Power-Inverter-Charger-p-9.html

get two large marine deep cycle batteries at Walmart and you'll have enough reserve 120VAC to run your pump for days during a blackout.

in fact I have two such systems each powering its own dedicated AC main pump. 2 primary pumps, 2 power supplies, 4 deep cycle batteries. redundancy is peace of mind. bring on the hurricane!

oh and forget the basement watchdog backup systems - they are junk. for many reasons. I threw mine out.

also as suggested make sure your downspouts are not spilling over near your foundation. that's probably the #1 cause for high water volume to your sump. you'll have to get out there during a big storm and check them in person.

holy lord that's a load of water. You sure your gutters are clear and downspouts are not exiting water next to your foundation?