It is not so novel to use a phrase such as “the prisoners were executed” when describing, for instance, a war crime. And there is a pattern in crime reporting of using the phrase “the victims were shot execution-style” to describe that it was done in cold blood with a non-resisting or restrained victim. So with time and loose application its common usage drifts into describing any killing done against someone who has been rendered helpless or captured, regardless of legality(*).
Things like that happen with language.
Same with “claim responsibility”; as mentioned, anyone and his brother in law can call and say they were behind a crime. When some people object to any word or phrase other than murder/guilt being used, they feel that it’s a euphemism or somehow it grants the criminals an undeserved dignity. Reminds me some years ago Fox News/NewsCorp tried to do away with the phrase “suicide bomber/bombing” and replace it with “homicide bomber/bombing”. They felt that it was improper to make the death of the bomber the defining characteristic of the incident; but of course the deal is that is what makes *that *kind of attack distinctive from other bombings that kill people. Didn’t catch up with anyone else.
(*And remember that in much of the Western world there is no such thing as a lawful execution any more)