Ignore the article you read. It's misleading.
Use 'a' if the 'second attempt' you are talking about has not happened yet. Use the if the 'second attempt' has already happened, and you are referring to it at a later point. Therefore, 'can the driver make a second attempt today?' is perfectly correct, because the driver is being asked to do something (the 'second attempt') that has not already happened.
If you were talking about the first attempt, which presumably failed, then you would use 'the', because if it failed, then it must have already happened (if you remember the rule I mentioned). So, you could say 'unfortunately, the driver failed on the first attempt', but NOT 'unfortunately, the driver failed on a first attempt'.
Similarly, you should NOT say 'can the driver make a second attempt today?' because the 'second attempt' is merely a REQUEST, meaning it obviously has not already happened.
Feel free to ask about this in the comment section, if you want me to make anything clearer.