Which rules govern what happens when a passenger doesn't board their flight with checked-in luggage?

There is no such thing as far as I know but it is hard prove a negative. Searching online, which I'm sure you have already done, does not seem to find such document. There is a regulation called Passenger Positive Bombing (PPBM) that which is well described in this answer but surprisingly hard to find the official document online.

These things depend on both country and local policies. I have seen cases when the baggage is off-loaded but not always and much less so in the last 10+ years. This could be because there is greater confidence in screening. A particular country can have more strict requirements which would dictate if the flight is allowed to take off with the baggage still on board.

Generally, and there is no official basis for this, airlines tend to be more forgiving with things when they are out of a passenger's control than otherwise. If you missed a connection because they were late, your luggage might be there already and they often do not off-load it. If you drop luggage off at baggage drop and don't show for a flight an hour or so later, there is greater cause for concern, so they are more likely to off-load.

There is no "international legal document" governing such cases. The document that governs what happens when you're late for a flight is the airline's Contract of Carriage. These typically require you to be present at the gate and ready to board at least 10 or 15 minutes prior to scheduled departure, though it may be more for international flights. Should you miss the flight, the contract of carriage typically specifies that you are considered a no-show and your ticket is forfeited, including any later flights on the same ticket. Airlines may reaccommodate you without charge as a courtesy, but they're typically under no obligation to do so. But, again, this would vary based on the CoC.

Positive Passenger Bag Matching (PPBM) laws in some countries may require your bag to be removed if you miss the flight for a reason under your own control, but these laws vary by country. Some countries (such as the USA) have no PPBM requirements, since all checked luggage is screened for explosives before it is allowed to be loaded on any aircraft.

In order to answer why PPBM didn't apply to your flight, you'd need to ask a new question and include the origin and destination of your missed flight. Mentioning the airline would also be helpful in determining what rights you may or may not have, since every airline has its own contract of carriage.

There is no international law. Every country might have its regulations or laws, and they might have strong similarities, but there is no overall organization that could make any international law.

For understanding your situation, consider why the luggage offloading rule exists: A passenger should not be able to get (potentially dangerous) luggage onto a flight if he is not himself on the flight - this assumes that most terrorists would prefer to survive their attack by not flying on the plane with their bomb.
If a passenger is not on a flight for reasons clearly out of his control, the offloading can be skipped - it would be quite disruptive for the schedule, and he could not intentionally have produced the situation.

Sorry, there is no such thing. And related rules and precedent you may find will heavy favor the airline. Some examples can be found at IATA - Baggage Standards.

Based on the question, you were late so the responsibility is solely on you.

Presumably, the connecting flight was domestic where bag match rules have more exceptions, though airlines will most often pull bags.

There is definitely an international standard, though it is vague:

ICAO Annex 17

4.5.3 Each Contracting State shall ensure that commercial air transport operators do not transport the baggage of persons who are not on board the aircraft unless that baggage is identified as unaccompanied and subjected to appropriate screening.

If this was the United States, 49 USC 44901 says that you either need to scan with approved explosives detection systems OR manual/dog inspection OR offload baggage (PPBM). The former 2 options would be "appropriate screening" as ICAO says.

The EU is similar:

EU EC 2320/2002

5.2.2 Unaccompanied Hold Baggage. All items of unaccompanied baggage, both originating and transfer hold baggage, shall be screened by one of the following methods, before being loaded onto an aircraft:

(a) EDS; or

(b) a multi-level PEDS, where at Level 2 the images of all bags are viewed by the operators; or

(c) conventional x-ray equipment with each bag being viewed from two different angles by the same operator at the same screening point; or

(d) hand search supplemented by the application of Trace Detection Equipment on open pieces of baggage,

unless the unaccompanied baggage, which has been previously screened to the standard detailed in this Annex, has been separated due to factors beyond the passenger's control, and the unaccompanied baggage has been within the care of the air carrier.