The initiative formula consists of the four categories described at 6.4, bottom of page 76 of rules (for army commander).
Initiative is calculated differently for each group of leaders: TC, AC, unit commander not in region with AC. AC is the most important and you use that as an example, so let me try to help on the AC initiative.
As you noted, the four categories are:
attack rating (modified)
home (or depot) modifier
"help" from the TC
"time" factor added by patch but not discussed in the rule book, it is found in the addendum
these four factors are added together and compared to a single die roll. The die has 14 faces in non-winter 1862 and after, 21 faces in non-winter 1861, and 28 faces in winter.
In your example the die roll was 2+5+5-2 = 10 divided by 21 = 47%. Think of this as a fraction. the four categories are the numerator and the season/year is the denominator. 2+5+5-2 divided by 21 = 47%. Leader has a 47% chance.
The number in the red circle is not your die roll, it is what your die roll is compared to. The red circle number changes according to the season/year. The "21" in your example means the leader is checking initiative during a non-winter turn in 1861. If it were November 1862 that number in the red circle would be 28; if June 1863 that number would be 14.
Hope this helps (and I hope I am correct).
"L'audace, l'audace, toujours l'audace."