What is the definition of an "objective kill"?

Based on experience you get an Objective Kill when you kill (or damage) an enemy player while he is on or near the objective.

Meaning that on payload maps an enemy player would need to be actively pushing or contesting the payload.

Same goes for point captures, where, to get an objective kill, an enemy would need to be on the point that is to capture or to defend.

This also means that you cannot get more Objective Kills than normal Kills.

From personal experience, you get Objective Kills when getting an elimination while standing on/next to the objective as well as if you help eliminate someone on the objective. Basically, it is an indication of how much you were fighting over the objective, whether defending it, attacking it, or pushing it.