FHE is the ability to encrypt some data and give it to a machine, which is then able to perform sensible operations on it without decrypting it (and thus ever learning the plaintext).

The term you are looking for is "obfuscation". Either black-box obfuscation (which was shown to be impossible) or the indistinguishability obfuscation, which is what the article talks about.

You can encrypt a function with FHE, but the applications of that are limited because the output is encrypted as well. We've had provably secure FHE since Gentry's 2009 paper.

Obfuscation roughly means "encrypting a function." But here be dragons: since the adversary gets non-black-box access to the implementation of the function (a circuit), and can also learn as many (input, output = f(input)) pairs as they want, it becomes impossible to say that the adversary cannot learn anything else about f in the general case (intuitively, consider a higher-order function f: you could learn a lot about f by computing f(f)).