In the USA, generally a documented vessel is one listed in the Federal Registry. This registry is kept by vessel name and no two vessels can have the same name. They can and do have variations on the same name, such as Rocky, Rocky One, Rocky I, Rocky Two, Rocky Too, and so one, however, the US government has the final say whether a similar name can be used.
A Registered or Licensed vessel usually refers to a US State method of registering vessels based in their state waters, in most cases to collect various taxes. Generally these vessels are assigned a unique number indicating the State of registry or licensing and a sequential number. For example, FL1234AB indicates Florida registered vessel with the sequential number 1234AB. These numbers must appear on the vessel.
In Canada, there are no documented vessels but vessels must be either Registered or Licensed. A Registered vessel is registered with the Federal government as a British ship and like the US Documented vessels use names as the distinguishing feature.
A Canadian licensed vessel is registered with Canadian Customs, or whatever name they go by at this time, and a number comprised of letters and digits identify the area of the country where originally licensed.
If the vessel does not plan to travel outside of the home country waters there is little point in Documenting a US vessel or Registering a Canadian vessel.
Check with your local registrar as there are exceptions to the rule and exemptions for smaller, unpowered or underpowered boats. They are the final authority.