If you're giving the numbers their original values and not using new labels then you can definitely start at any number you like. You could even start at 0 but in that case you won't be referring to a finger that's there, but nothing. If you're starting at 2 then you would be counting 2 fingers that are there, providing there are 2 there to count.

You don't have to count 2 twice because you're counting in whole numbers, which are in increments of 1. You don't use the same logic as 0 because nothing doesn't have to be counted. That doesn't mean you can't or don't have to start counting at 0 but that most people will skip counting nothing because all they care about is the somethings they're counting.

If you count 2 twice then you would have 4 and might be counting by 2s instead of 1s. If you count 3 three times then you'll have 9 and might be counting by 3s instead of 1s. If you counted 2 twice or 3 thrice and so on then you'd be counting more than necessary. Counting something isn't the same as counting nothing because there is something there to count. Although you could count nothing as many times as you like but you'd still have nothing. You could divide those nothings up into multiple nothings but that's not going to make them something, they'll still be nothing. If you start counting spaces where something could be then in that case you are no longer counting nothing, you're counting spaces. 0 represents having no finger there which is why it doesn't have to be counted while 2 or 3 represents 2 or 3 fingers being there and so they can be counted.

I don't think the number you start counting at has to be arbitrary unless maybe you want it to be or maybe if you are giving them different labels or values. It's not going to change how many there actually are no matter how you count them but you you might come up with different results if you aren't counting them properly or if you're using your own labels to describe the numbers.