If in doubt, consider it a new cycle, but consider yourself potentially fertile during bleeding. A true period will be preceeded by ovulation, confirmed by CM dry up cross-checked by a temperature rise. Without this, you can't really know for sure if you have ovulated or not. Breakthrough bleeding can occur mid-cycle that is usually much lighter and lasts a shorter time, but breakthrough bleeding for one person may look like a period to another, so unless a woman is really familiar with her cycles, it's hard to tell. It often occurs shortly before ovulation, which is why I say to consider yourself potentially fertile, but can occur at any point in the cycle.
Anovulatory cycles are more common if your fertility is just returning while breastfeeding or after coming off a long term birth control method or if you have PCOS or certain other medical conditions but can happen on occasion in normal women. That is where you don't ovulate, but you have 2 sets of bleeding a few weeks apart that both look a lot like a period. What you're describing sounds more like breakthrough bleeding than an anovulatory cycle, but if any of the above conditions apply to you, they might explain why this is happening.
In any case, I think it's highly unlikely that those were both real periods, and it sounds like the second one was more likely to be a real period.