There’s been a lot of talk about whether or not an ex-US Coast Guard cutter — like the BRP Gregorio del Pilar and BRP Ramon Alcaraz, can really be called “warships”. Even among individuals responsible for government communication disagreements reported exist.
When looking for internationally accepted definitions, the United Nations is a reasonably good reference. For maritime matters, there is UNCLOS. Article 29 of UNCLOS provides a reasonable definition for what constitutes a warship:
Definition of warships
For the purposes of this Convention, “warship” means a ship belonging to the armed forces of a State bearing the external marks distinguishing such ships of its nationality, under the command of an officer duly commissioned by the government of the State and whose name appears in the appropriate service list or its equivalent, and manned by a crew which is under regular armed forces discipline.
Note that the article does not consider the vessel’s armament. The key criteria are ship ownership and the nature of the crew. So technically speaking, even an unarmed tug is a warship.