What are the leading FHE schemes? I'm trying to understand the current context for fully homomorphic encryption. What are the most influential papers that are the basis for the most close-to-practical techniques today?

Of course Gentry's original paper is the most influential, but it is far from practical. A 2011 paper I see that appears to achieve much better performance (also by Gentry) is https://eprint.iacr.org/2011/277.pdf. What other papers published in the last 7 years are worth reading? (I would prefer papers with a post-quantum context in mind).
In short

BGV (the one you linked), FV and GSW.

More details

The scheme you have linked is known as BGV and I would say that it is the most important. There is a public implementation on Github.
If the application can run in parallel, then BGV is probably the best current choice, because its plaintext space can be view as a product of rings, which gives the user the ability to encrypt several values in a single ciphertext. Then, a single homomorphic operation is actually equivalent to several plaintext operation in parallel.

The FV is another scheme with that feature and is the one that SEAL implements now.

If the application can't take advantage of the parallelism, then, I think the best choice nowadays is the so called FHE over the Torus, or TFHE.

Since you asked for influential papers, it is worth noting that TFHE is based on FHEW, by Miccianco, which is based on a paper by Sheriff Peikert, which is based on (this highly influential paper) GSW, again by Gentry.

Also, YASHE had an efficient version that was implemented by SEAL before, but it turned out to be insecure.