Dimensions are different for an internal thread where you use a tap and an external thread where you use a die. Internal thread you need to look at the upper and lower limits for the minor diameter under internal threads. External threads where you use a die you want to look at the major diameter for an external thread.
So cutting a external thread for a 10MX1.25 with a die, you would still want to start with an OD of around 9.9". The major diameter range varies slightly based on the tolerance class for the particular thread, just realize that there needs to be some degree of clearance between the internal and external thread. If I was cutting an external thread I probably would start out with a 10 mm rod and thread it on the lathe and then take off the peaks with a file, and/or chase it with an adjustable die for final sizing.
Cutting internal threads with a tap is based on the minor thread diameter which changes with the thread pitch. A M10x1.25 the minor diameter range is 8.647-8.912 mm, the tapping chart recommends using a 8.70 mm or 0.3425" drill size. I would use a lettered drill "S" which is 0.3480". The Major diameter that the tap cuts should be slightly over 10 mm The major diameter range is 10.000-10.340". The degree of thread forming may also vary based on the material, the size of the tap and the strength needed. In some materials with smaller tabs you may want to start with a slightly larger hole to prevent the tap from breaking, assume you do not need maximum strength of the thread.
There are slight variations on the dimensions depending on the charts you use and the tap/dies you use.