Which are the "Light Machine Guns" they are referring to here?
Any gun in the Support class counts as an LMG:
Note This does not include guns that the support can use, only guns that are exclusive to the support class. So any multi class gun is not a LMG. Just because support can use it doesn't make it an LMG. Some guns that don't count:
Looks to me by your battle-log that you only have the M249, so spawn Support with that and rack up those kills!