The product you link to sounds like something that fell off a factory truck in China. The price just seems too low for that much power. Something has to give, somewhere. That said, it's not much risk for under $30. 1000 lumens is WAY more than you need for commuting unless you are on rough trails where you need to see obstacles at a distance. Even then, you generally want 2 lights so you can see the 3 dimensions of the terrain better without as many strange shadows.
I would personally rather spend more on a more user-friendly headlight and get a cheaper rear blinky.
It slightly exceeds your total cost criteria, but... For the front, something
like this would be more than adequate, much more convenient, and with better customer service than the one you linked to. At $65, you could get this and add a rear blinky for under $10 and be close to your budget with what is probably more reliable. If you take your light on/off a lot, if you are at a store, etc. it's nice not to deal with cords and battery packs.
I have the 650 lumen version and run it on low on dark streets and wide gravel trails. I also have a 150 lumen helmet mounted headlight. I like having the backup, and add in the extra light from the helmet when on trails. If you only have one light, handlebar is safer than helmet.
On back, the cheapest blinky you can find, along with a red reflector for backup is probably fine. I personally prefer the planet bike super flash as a rear blinker. It's quite bright, but doesn't double as a reflector if your batteries die,
