Blockchains have been overloaded by a bunch of people bringing modern technology to ancient industries as a sort of buzzword. So to an extent you are right the author's definition is narrow, and on the other hand the author is right in that their narrow definition is the correct one.
Blockchains are generally defined by the original paper for bitcoin: https://bitcoin.org/bitcoin.pdf these would have a couple of key properties. They are trustless, distributed, and public through the use of cryptographic and computational algorithms. Most modern "blockchains" (that aren't currencies of some form) violate one of these properties.
> When I think "blockchain", I basically think of a git branch: history is verifiably immutable and using some external tools like a public rng, one can prove that a block was not created before a certain time.
The system you describe here involves trust. "Public RNG" implies we are talking about some sort of SSL certificate company's time server. Proof of work is stronger than that. It says it's impossible for someone to misrepresent the shared ledger unless they have more computational power than the entire system put together. And it's a fundamental part of how the ledger is structured (it's not external).
> When I think "distributed ledger/distributed consensus" I think of the practice of using basically properties of statistics to get a bunch of nodes to agree on the state of something.
Again proof of work is stronger than this. The nodes agree because the given branch is provably the strongest branch of truth. Distributed consensus is about getting the nodes to work together when you can trust they all want to, not when they are all trying to compete with each other.
> Neither of those things really seem to make sense as the author's definition of "blockchain". What is he referring to?
Because neither of your definitions fit a blockchain. Your definitions fit the broader cases of technology being passed off as "blockchain" in (the best light) an attempt to get these old companies to modernize (in a worse light) and take their money because they are idiots.
Things like distributed version control systems, distrusted consensus systems, and cryptographic ledgers are all useful, but they involve trust. And for a lot of these applications that's fine. A company putting out a cryptographic ledger (e.g. provable) for cosmetic item transactions is an improvement, but it's not a block-chain (it's not distributed or it's not trustless).
I mean the author discusses this: "Be careful! Today, digital signatures are often sold as blockchain. Perhaps a digital signature is all you really need and blockchain does not really suit your requirements." a digital signature system of arbitrary complexity (perhaps with a ledger of issued signatures, or an external time server, or agreement on which signature is the winner) is all most people in this space need.
The only reason one needs a block chain is if they are making a currency.