You may persist with the idea that this is SAR only, but that is a semantic distinction only
Not at all. The three would be led by very differing government bodies (with potentially differing levels of competence, and perhaps hidden agendas)
SAR is usually under the command and responsibility of the organization owning the principal SAR assets used (in this case primarily Malaysian armed forces).
Air accident investigation is conducted by the responsible (and supposedly independent) transportation accident investigation body, and a part of this investigation is to assess the quality of SAR activities. It is launched immediately after an accident becomes known (and may in fact run parallel to SAR).
A criminal investigation is run by the judicial system, triggered by the 'discovery' of a criminal act.
Not quite semantics here...
Potentially the biggest dilemma if the aircraft is not found is whether this should be treated as an accident investigation or criminal investigation. Depending on the path chosen, some evidence may be easier to gather, but some more difficult. (Eg. in a criminal investigation any request for evidence from another country must pass through judicial and diplomatic channels, while in an air accident investigation various foreign investigation bodies may be invited to form a part of the investigation team).