What 'counts' as a concussion is still an open discussion in the medical community. It's been known for a while that getting knocked unconscious regularly leads to brain impairment (Dementia Pugilistica). It was thought for a long time that if you weren't getting knocked out, you were ok - however the data from the NFL/Iraq War/etc. is that even sub loss of consciousness hits can cumulatively lead to problems down the road (Chromatic Traumatic Encephalopathy). The recommendation to not get multiple concussions in a row is based on something completely different - that the headaches and difficulty concentrating tend to stick around for longer the more concussions you've recently had. The extreme version of that is Second Impact Syndrome - which is really rare, but you also really don't want it to happen to you.
We're taught in medical school that anything that causes a change in your thinking - a headache, dizziness, etc - counts as a concussion, which is synonymous with the term 'mild TBI'. Those are graded from minor (briefly feeling dizzy etc.) to more serious (loss of consciousness for 5+ minutes, severe memory loss). Do minor concussions matter down the road? Nobody really knows. All that's clear is that getting a bunch of severe concussions is bad, and that minor ones may be worse than originally thought.