Check out this thread:
https://forums.macrumors.com/threads/owc-or-apple-replacement-battery.2108502/
It talks about batteries for MacBook Pro, but I think a lot is applicable to the Air.
What I take from it:
1) People have mixed reviews about EBay batteries.
2) Apple won't do anything with your machine if it's "vintage" (more than 5 years old). Otherwise, I'd probably try them at $129.
3) The thread starter had decent success with a NuPower battery.
And I agree about Amazon. The reviews are sketchy. One of the batteries on there had a bunch of positive reviews, all dated 12/4. And one (maybe the same one?) supposedly was thicker than the OEM battery and causes the touch pad to be hard to press, etc.
I too have a MacBook Air A1466 (mid 2012), that I purchased in March 2013. And my battery needs replacement.
Based on that thread above, today I ordered the NuPower battery at B&H (it's also available at OWC):
https://www.bhphotovideo.com/bnh/controller/home?O=invoice&A=details&Q=&sku=1219802&is=REG
It has good reviews. It's a little pricier than the ones on Amazon, but I guess we'll see. B&H and OWC are supposedly pretty respectable.
BTW, I also ordered a new power cable on OWC the other day. It's supposedly an OEM part, and it's cheaper than buying from Apple. My cable has split several times, and I keep taping it up. I wish they were more durable, but Apple cables are notorious for this.
https://eshop.macsales.com/item/Apple/MD592LLA/