ANY bleeding that you have on the pill that is not on your placebo week is called breakthrough bleeding, it is not a period.
When a doctor asks you "When was your last 'period'?" He wants to know your placebo week, not any other bleeding in between, because that is not your period.
You should keep taking the pill as usual, and you will have your period during the placebo week.
When women start or restart the pill, they tend to have odd patterns of bleeding for the first month.