The construct "ich war laufen" / "Ich bin laufen gegangen" is, as RHa pointed out correctly, called Absentiv.
Both your examples are not so much an answer to "what did you do?" but rather to "where have you been?"
You were asked for a location and answer with an activity (which may somehow imply a location - If you answer "ich war schwimmen" that closely implies you might have been somewhere where this is possible, probably to the pool). That activity is your reasoning why you have been absent, thus Absentiv.
The difference between your two examples is mainly that
Ich war laufen
would answer the question "where have you been?" and
Ich bin laufen gegangen
would answer the question "where did you go?"
The Absentiv is always constructed with some form of "sein" + infinitive, regardless of whether the verb would rather take "haben" as auxiliary in perfect tense:
Ich war arbeiten
Means "I was at work/in the office/workshop/...", even if arbeiten requires "haben" as auxiliary in perfect tense.