RPC server with bitcoin-qt and bitcoind has a different set of commands?
Just discovered the problem. It has to do with the "wallet" being disabled in the code via the default configuration with my package management.

I investigated the source initially and saw that there is a bunch of commands that are only enabled if the wallet #define is enabled: https://github.com/bitcoin/bitcoin/blob/e18bc583d41d97cdab93b7ffd111402628cfba19/src/rpcserver.cpp#L268

From there I was able to work out that the build options has disabled the wallet. I'm on Arch Linux, which the user-contributed package (on the AUR) has been configured to disable the wallet by default.

In other words, PKGBUILD configuration has added --disable-wallet to the  ./configure arguments as you can see here: https://aur.archlinux.org/packages/bi/bitcoin-core/PKGBUILD

As if the time of posting, the ./configure command is:

./configure \
    --prefix=/usr \
    --sbindir=/usr/bin \
    --libexecdir=/usr/lib/bitcoin \
    --sysconfdir=/etc \
    --sharedstatedir=/usr/share/bitcoin \
    --localstatedir=/var/lib/bitcoin \
    --enable-hardening \
    --with-gui=no \
    --disable-wallet


Modifying PKGBUILD by hand to remove the --disable-wallet argument fixes the problem.