Using the most broad common definition of the terms, a domestic flight is a flight entirely within one country, while an international flight is a flight between two countries. Therefore, there is no such thing as a domestic flight between two countries.
But whether a flight is international or domestic in this sense is really just a matter of nomenclature, one that relies a lot on the often unclear question of what is a country. (Is a flight between London and Edinburgh a flight between two countries?) What really matters to a traveler is not the terminology, but what rules and procedures apply to the trip, such as:
- What travel documents are required (passport, visa, identification, special travel permit, etc...)
- What inspection formalities (immigration, customs, agriculture, etc...) apply and where are they conducted?
- Does the flight depart from or arrive at a separate "international terminal?" This can be misleading: in some cases, domestic flights may leave from an international terminal and vice versa to make the most efficient use of airport resources.
- What airline policies apply for minimum check-in time, baggage limits and fees, meal services, availability of services like lie-flat seats in premium cabins, etc... These may vary by domestic/international status, but also the length of the flight and the specific city pairs involved.
And the answer to these questions depends on the specific destinations involved. To a first-order approximation, international flights within the Schengen area feel like domestic flights to travelers, as there are no immigration controls. Conversely, some flights that are technically domestic flights might require immigration controls, typically involving territories (e.g. Beijing to/from Hong Kong). Other international flights may involve Preclearence at the destination airport, which typically requires an earlier check-in time, so they may "arrive as domestic flights" at their destination (without further immigration controls). Still other flights may be entirely domestic, but require special formalities, such as government permission to enter a specific area or an agricultural inspection (e.g. when flying from Hawaii to the US Mainland).
The specific case you've mentioned is one where both the ROC and PRC find it in their general interest to maintain that a flight from Taipai to, say, Beijing is a domestic flight, even though the two are controlled by two separate governments. The governments involved officially consider the territory a single country and so will call such a flight "domestic," but if you want to call it "international" in the privacy of your own home (because they sure seem like two countries if you don't know anything about the history and politics), feel free, as long as you don't care what the Chinese government thinks of you. Again, the relevant bit for travelers is what policies apply to Cross-Strait flights, and that tends to involve special travel documents like the Mainland Travel Permit and the Entry & Exit Permit.