US airlines require line checks at given intervals. This checker is called a “Check Airmain” in the FAA terminology. JAR-OPS has a similar requirement for line-checks. Question to you guy’s working for various EU carriers: What does your company call the person administering a line check? (in addition to the fact that some of them also might also be TRI/TRE)
There is no specific title although they must complete a company course, and their names be approved by the licensing authority. Line checking duties are often carried out by TRI/TREs in smaller companies. Line checks in BA are called Route Checks (they do like to be different), and they have specific 'Route Check Captains' who do not take part in training duties.