That is normal to have that kind of spotting since you missed two pills..
Any bleeding that is not on your placebo week, even if it is heavy, is not considered your period, just breakthorugh bleeding.
When you miss 2 pills, you should take the two pills as soon as you remember,( even if you take more than 1 on a day) and keep taking the rest as usual..Basically you play catch up..Then you need a back-up for 7 days..
You did the right thing in starting a new pack , but you should not have waited for the bleeding to stop.. If you miss 3 or more pills then you would definately start a new pack, but with 2 you do not have to.
Anytime you are not on placebo pills, even if you miss a pill or are bleeding, you should keep taking a pill every day.