5 - 10 = 5 + (-10) = 0x05 + (0xF5 + 1) = 0x05 + 0xF6 = 0xFB
00000101 -- 0x05
11110101 -- 0xF5
+ 00000001 -- 0x01
==========
11111011 -- 0xFB
And this continues for 16 or 32 or 64 bits 0+1+0 = 1, carry 0
You are right in the sense that it doesnt carry out. A subtract is an add with the second operand inverted and the carry in inverted. Some processor families invert the carry out some dont. Sounds like you are looking at something that inverts the carry out on a subtract.
So if 5 - 10 gives carry out (borrow) then try 10 - 5 on the same processor, you should not see the carry bit set (no borrow).
Your second example is an add operation, the carry out is not inverted on any processor that I know of, further supporting the carry bit being positive logic indicating a borrow.