Miles and More Number vs Service Card Number
The 15-digit number is the service card number (the number shown on your M&M card).
The 9-digit number shown in your account after login is your customer number.
Personnally I always use the 15-digit number in communications with M&M. This number is also the one shown (partially) on your boarding passes.
I never used the customer number, but I could see that it was used by OS when they did reply to me via email when I did contact them via online form on the internet site.