I know that gets tricky, and I assumed you needed to know the "exact" date for a reason! I had my first very strange period as you describe when I was TTC (one entire week of spotting, followed by another week of "normal" period)...and of course, that was the last period before I actually did conceive. I just told my OB/GYN both dates, and we went from there. That was 2 years ago, and I believe she wrote down the first date (the spotting date) as my LMP..just one woman's experience...your's could be different!
Just be sure to write both dates down, and if they are just a few days apart, I assume they would go with the first date of spotting.