purchase apple care within a year

Depends on where you are in the world and what device it is for, as explained in the support document you posted from:

• In most countries and regions, you can buy AppleCare+ within 60 days of your device purchase. In Japan, you can buy AppleCare+ within 30 days of your device purchase. In China mainland, you can buy AppleCare+ on your device within 7 days of your device purchase or in an Apple Store within 60 days of your device purchase. (At the Apple Store, you'll need to bring your proof of purchase, and we'll inspect the device.)
• In some countries and regions, you can buy an AppleCare Protection Plan within 1 year of your purchase of an eligible Mac, Apple TV, or other device.