Ankou has already given you very good advice here and correctly identified two of the factors that drive the risk of contracting an asbestos disease up. The third one is the type of asbestos. Without getting too far into technical details, the 3 types are commonly known as blue white and brown. The vast majority of asbestos cement is made up of about 15% amosite - brown asbestos which is the least aggressive. Coupled with the fact that only a small percentage of the material you identified is asbestos, the risk is very small.
TB is correct that there has been no asbestos in these type of products for many years now (can't remember how many - at a guess, 25 years) but most asbestos cement products being removed date back to the 50s, 60s and 70s - when the greatest volume use of their use occurred.
You can go to your doctor and explain the situation - I did this for similar reasons to you and was offered a chest X-ray. As you were young at the time, I would do this because younger lungs have been shown to show as increased risk (only marginally though).
TB - the brand 'Eternit' is a product brand owned by Marley. Eternit used to contain a low percentage of amosite, so it is not true that it is asbestos-free. Black amosite toilet cisterns and lino tiles are 2 sources of this - it is OK to remove these products if in non-friable condition provided one doesn't drill into them, but disposal requires a proper licensed disposal.
I find it unhelpful that deggers mentions his relative in the context of this question. I guess we all know an unfortunate person who has succumbed to this. Asbestos used for insulation purposes (lagging) is the worse issue, and blue asbestos was often used here - the fibres break off into free air - to give an idea the risk factor was perhaps 500x more than what happened to you. But as is so often the case, news sources just love to give the worst scenario when reporting risk - it sells newspapers.