Spelger, you correctly say “if the bike computer has the correct information”. IF!!!
I have always been a little suspicious about using “standard” conversion tables for translating wheel “size” to wheel circumference. (eg 700 x 23C = 25-622 = 2096)

My general knowledge tells me that standard wheel size will only equal distance travelled for any given tyre pressure or even tyre wear. A flatter, less inflated 700x23C will have a smaller circumference compared to a well/highly inflated 700x23C. Tyre wear and even manufacturing tolerances (exact amount of rubber all the way around a tyre) would affect circumference in a similar, albeit to a smaller extent.

Therefore, using 2096 in a BC assumes that your actual wheel on your bike rolls 2096mm for every revolution. Thus, using standard conversion table circumferences is assuming your BC is using the “correct” information. To me, that is not necessarily a good assumption and necessitated some sort of recording and comparison.