What Characteristics Define a Rudder ?

I wouldn't push that too hard in SE Asia and the SW Pacific. A fair number of indigenous small craft on the high seas. One of the reasons the PRC wishes to "help" the Solomon Islands with their maritime "policing" issues.

 

Rudders have nothing to do with registration. Any form of boat document is subject to national regulations, each country decides individually. What this means is that the need for boat documents depends on three things, your own nationality, your residency and your intended area of cruising.

There are many countries where boats (and mariners for that matter) don't have to have any form of documents whatsoever. This applies wholesale or to certain types of boat. The situation is fine for domestic waters and all other areas that accept it. Now if you intend to go to a country where a lack of documents is not acceptable, you need to obtain them, otherwise you are not getting in.

The UK is a good example, recreational boats don't need any documents. This worked fine until France said that its not acceptable anymore and british yachts had to have some papers. Instead of subjecting yachts to a full commercial registration the UK created the SSR, a dedicated yacht register wich proves nationality but not ownership.
A more recent example is the dutch "light" registration, wich doesn't provide nationality and while perfectly valid in The Netherlands as ownership proof, is not accepted anymore in Europe.
Current USA interpretation is that only federal documentation provides vessel nationality, state registered boats take their nationality from the owner.

In order to cross borders just as you need a passport and sometimes a visa, your boat needs the same (visa beeing temporary import permit, cruising licence, etc.). What is acceptable depends on the regulations of the country you want to visit and how well the people there understand the papers you present. Think of it like this, just as there are countries where you don't get in without a biometric passport, there are countries wich want specific papers for your boat.
This papers can be the same ones commercial boats get, and there are many countries where this is the only option. Such papers do have a degree of standardization regarding the information they provide and are universally accepted. Then there are all other forms of registration countries provide to yachts, and some of them are not acceptable in some countries.

As a US citizen you should get federal documentation. State registration is acceptable in many jurisdictions but not actually intended for international travel. If for some reasons you desire a flag of convenience, you have many options, from open registries, to some that specifically allow US citizens (UK part 1 for example), and since you are an expat the registry of the country of your residence.

 

I only know of this distinction from a very long time ago on the Mississippi and it's tributaries. There was a distinction between boats with a rudder and boats without. It was based on the assumption that a boat with a rudder had a "captain" whereas row boats and paddle craft without did not, and it was therefore more difficult to hold someone accountable in the event of a problem. There has been a huge effort to figure out what sort of craft were used by Lewis and Clark, without much real success. But it seems they bought a couple of boats that had rudders for their trip down to St Louis, possibly for the sole purpose of being taken seriously and being unequivocally the people in charge of the expedition. There were different names for identically constructed boats that did and did not possess rudders. They appear have been a bit of a prestige item which gave the craft's operator some legitimacy.

 

That too, but also for accountability if the boat is abandoned. Especially so for motored vehicles as they can pollute the water.

 

Actually registration of small craft in the US is a Federal requirement that States register certain types of boats. All this can be found in Title 33 Of the Code of Federal regulations specifically parts 173 and 174. I won't get into the weeds on this but what has happened since that regulation was enacted is that each state has decide that they each have different requirements above and beyond the Federal, which started long ago with boats having mechanical propulsion of 10 HP or more, but now applies to any boat with any means of mechanical propulsion.
33 CFR 173.11 Applicability says in part;
"This subpart applies to each vessel equipped with propulsion machinery of any type used on waters subject to the jurisdiction of the United States and on the high seas beyond the territorial seas for vessels owned in the United States except (and there is a laundry list of boats that don't have to register, mostly publicly owned boats )

Thus the proliferation of outboards rated for 9.9, many of which do develop far more than 9.9. This has become an industry norm, and still applies in many states on waters where the State has sole jurisidiction. This why my 12 foot rowboat which is rated for 2 hp has to be registered, because the lake where I use it is joint Federal/state jurisdiction. If it were sole state I wouldn't have to register it. So it is incumbent on the boat owner to know what the requirements are in their own state. The Federal regulation also says that the boat must be registered in the state of "principle use". Most states have determined this to be 90 days or more, but a few use 60 days or more.

But I'll cut this short by stating that having a rudder has absolutely nothing to do with it.