Why make this function impure by passing a Customer object? I would extract everything but the assignment into a pure static function.

Second, it seems some corner cases are uncovered? What happens if discount > orderTotal? Do you send customer some money? Maybe it’s an exception?

Finally, I think Sum will return zero for an empty list.

I would probably rewrite as:
{
decimal orderTotal = products.Sum(p => p.Price);
return Math.Max(0, orderTotal – discount);
}

Udit: I would do begin and end of function logging as in the caller or with a using() block.

(and again the bahavior in the two examples is different, because the Customer field may not be updated in the second example)

Why not do this?:
Customer = customer; // assuming the first example was the correct one
return products.Sum(p => p.Price) – discounts;