It's being used in a more literal-meanings-of-the-words sense ("pursuing monopoly rents") rather than the narrow economic term-of-art sense of "pursuing monopoly rents through influence over public policy by means that do not create, or which inhibit the creation of, additional wealth" (the definition you seem to be complaining about it not adhering to without actually providing.)
But most of the usages would also be correct in the narrower sense, because virtually ever actor referred to as rent-seeking in the broader sense are also rent-seeking in the narrow sense as part of that. (E.g., actively lobbying for "safety" regulation which would disproportionately impair non-incumbent new competitors.)
> What the author is talking about isn't rent-seeking per se but a moat.
Pursuing a moat is just another term for seeking monopoly rents by any means, including rent-seeking in the narrow sense.
(There's also obviously an ideological angle in creating the term "rent seeking" as a term of criticism to those seeking monopoly rents through means that the creators of the term disapprove of, excluding seeking the same kind of rents by other means from "rent seeking".)
Rent-seeking is fundamentally intermediation like a health insurer putting themselves between a patient and a healthcare provider or privatizing grazing lands or controlling water supplies from snowmelt (like the Resnicks) or privatizing trains in the UK.
Microsoft has a compeitive advantage with Windows, Google with its search engine and the ad business that funds it, Amazon with AWS or NVidia with GPUs. There are alternatives to all of these things but these companies maintain dominance with a combination of scale, cost, technology and network effects. That's not rent-seeking in a broad or narrow sense.
Rent-seeking would be Microsoft lobbying lawmakers to require schools and governments to purchase Windows, for example.
No, a moat is a barrier to people closing the gap of an existing competitive advantage. The barrier is literally the point of the metaphor. If there is no barrier, there is no moat, just a transitory advantage; that's the whole basis of the famous "we have no moat and neither does OpenAI" memo at Google; its not that the big established players had (at that time, or now) no current competitive advantage over open source competition, its that there was no structual barrier preserving it.
> Rent-seeking is fundamentally intermediation
No, rent-seeking is using influence over public policy as a means to extracting monopoly rents. State policy favoring intermediation is one manner of rent seeking, yes.
> Rent-seeking would be Microsoft lobbying lawmakers to require schools and governments to purchase Windows, for example.
That is an example, but lobbying for specific mandates for purchases from the specific firm is a fairly extreme case of rent-seeking, not the general case; a more common for of rent-seeking is firms in a narrow group of leading incumbents lobbying for various standards (either for public market access or government contracts) to be set for the industry that are not a particular burden for existing large firms to meet but which create additional friction for new competitors. In the specific case of software, this can include lobbying for rules that specifically treat open source solutions as suspect and dangerous, as well.
A moat can very much be transitory and caused by natural (or at least, not specifically intended) factors. Perhaps we could then call it something different, like a river as opposed to a moat, but the strategic effect is the same either way so it makes sense to use the established term for it.