REST API design for a "reset to defaults" operation

Let's discuss your mentioned options first:

• 1: DELETE does not necessarily need to delete or remove the state contained in the resource targeted by the URI. It just requires that the mapping of target URI to the resource is removed, which means that a consecutive request on the same URI should not return the state of the resource further, if no other operation was performed on that URI in the meantime. As you want to reuse the URI pointing to the client's settings resource, this is probably not the correct approch.
• 2: REST doesn't care about the spelling of the URI as long as it is valid according to RFC3986. There is no such thing as RESTful or RESTless URI. The URI as a whole is a pointer to a resource and a client should refrain from extracting knowledge of it by parsing and interpreting it. Client and server should though make use of link relation names URIs are attached to. This way URIs can be changed anytime and client will remain to be able to interact with the service further. The presented URI however leaves an RPC kind of smell, which an automated client is totally unaware of.
• 3: PATCH is actually pretty-similar to patching done by code versioning tools. Here a client should precalculate the steps needed to transform a source document to its desired form and contain these instructions into a so called patch document. If this patch document is applied by someone with the state of a document that matches the version used by the patch document, the changes should be applied correctly. In any other cases the outcome is uncertain. While application/json-patch+json is very similar to the philosophy on a patch-document containing separate instructions, application/merge-patch+json has a slightly different take on it by defining default rules (nulling out a property will lead to a removal, including a property will lead to its adding or update and leaving out properties will ignore these properties in the original document)
• 4: In this sense first retrieving the latest state from a resource and locally updating it/calculating the changes and then send the outcome to the server is probably the best approach of the ones listed. Here you should make use of conditional requests to guarantee that the changes are only applied on the version you recently downloaded and prevent issues by ruling out any intermediary changes done to that resource.

Basically, in a REST architecture the server offers a bunch of choices to a client that based on his task will chose one of the options and issue a request to the attached URI. Usually, the client is taught everything it needs to know by the server via form representations such as HTML forms, HAL forms or ION.

In such an environment settings is, as you mentioned, a valid resource on its own, so is also a default settings resource. So, in order to allow a client to reset his settings it is just a matter of "copying" the content of the default settings resource to the target settings resource. If you want to be WebDAV compliant, which is just an extension of HTTP, you could use the COPY HTTP operation (also see other registered HTTP operations at IANA). For plain HTTP clients though you might need a different approach so that any arbitrary HTTP clients will be able to reset settings to a desired default one.

How a server wants a client to perform that request can be taught via above mentioned form support. A very simplistic approach on the Web would be to send the client a HTML page with the settings pre-filled into the HTML form, maybe also allow the user to tweak his settings to his wishes beforehand, and then click a submit button to send the request to the URI present in the action attribute of the form, which can be any URI the server wants. As HTML only supports POST and GET in forms, on the Web you are restricted to POST.

One might think that just sending a payload containing the URI of the settings resource to reset and optionally the URI to the default settings to a dedicated endpoint via POST is enough and then let it perform its magic to reset the state to the default one. However, this approach does bypass caches and might let them believe that the old state is still valid. Caching in HTTP works as such that the de-facto URI of a resource is used as key and any non-safe operations performed on that URI will lead to an eviction of that stored content so that any consecutive requests would directly go to the server instead of being served by the cache instead. As you send the unsafe POSTrequest to a dedicated resource (or endpoint in terms of RPC) you miss out on the capability to inform the cache about the modification of the actual settings resource.

As REST is just a generalization of the interaction model used on the human Web, it is no miracle that the same concepts used on the Web also apply onto the application domain level. While you can use HTML here as well, JSON-based formats such as application/hal+json or the above mentioned HAL forms or ION formats are probably more popular. In general, the more media-type your service is able to support, the more likely the server will be to server a multitude of clients.

In contrast to the human Web, where images, buttons and further stuff provide an affordance of the respective control to a user, arbitrary clients, especially automated ones, usually don't coop with such affordances good. As such other ways to hint a client on the purpose of a URI or control element need to be provided, such as link relation names. While <<, <, >, >> may be used on a HTML page link to indicate first, previous, next and last elements in a collection, link relation here provide first, prev, next and last as alternatives. Such link relations should of course be either registered with IANA or at least follow the Web linking extension approach. A client looking up the URI on a prev relation will know the purpose of the URI as well as still be able to interact with the server if the URI ever changes. This is in essence also what HATEOAS is all about, using given controls to navigate the application though the state machine offered by the server.

Some general rules of thumb in designing applications for REST architectures are:

• Design the interaction as if you'd interact with a Web page on the human Web, or more formally as a state machine or domain application protocol, as Jim Webber termed it, a client can run through
• Let servers teach clients on how requests need to look like via support of different form types
• APIs shouldn't use typed resources but instead rely on content type negotiation
• The more media type your API or client supports the more likely it will be to interact with other peers

Long story short, in summary, a very basic approach is to offer a client a pre-filled form with all the data that makes up the default settings. The target URI of the action property targets the actual resource and thus also informs caches about the modification. This approach is on top also future-proof that clients will be served automatically with the new structure and properties a resource supports.

... so the audience and thus the potential types of "clients" I may encounter is rather small - developers who use my product, that is deployed in their private data center or AWS EC2 machines, and need to write a script in whatever language to automate some task rather than doing it via UI.

REST in the sense of Fielding's architectural style shines when there are a multitude of different clients interacting with your application and when there needs to be support for future evolution inherently integrated into the design. REST just gives you the flexibility to add new features down the road and well-behaved REST clients will just pick them up and continue. If you are either only interacting with a very limited set of clients, especially ones under your control, of if the likelihood of future changes are very small, REST might be overkill and not justify the additional overhead caused by the careful desing and implementation.

... some technical considerations like caching are relevant. Human user considerations, like how consistent the API design is across various resources, and how easy it is to learn, are also relevant. But "can some 3rd party crawler identify the next actions it can perform from a given state" isn't so relevant ...

The term API design already indicates that a more RPC-like approach is desired where certain operations are exposed user can invoke to perform some tasks. This is all fine as long as you don't call it REST API from Fielding's standpoint. The plain truth here is that there are hardly any applications/systems out there that really follow the REST architectural style but there are tons of "bad examples" who misuse the term REST and therefore indicate a wrong picture of the REST architecture, its purpose as well as its benefits and weaknesses. This is to some part a problem caused by people not reading Fielding's thesis (carefully) and partly due to the overall perference towards pragmatism and using/implementing shortcuts to get the job done ASAP.

In regards to the pragmatic take on "REST" it is hard to give an exact answer as everyone seems to understand different things about it. Most of those APIs rely on external documentation anyway, such as Swagger, OpenAPI and what not and here the URI seems to be the thing to give developers clue about the purpose. So a URI ending with .../settings/reset should be clear to most of the developers. Whether the URI has an RPC-smell to it or whether or not to follow the semantics of the respective HTTP operations, i.e. partial PUT or payloads within GET, is your design choice which you should document.