It sounds like someone missed the day they taught management in management school.
If these employees are paid hourly, THEY should be tracking their own arrival and departure by using an electronic time clock. You have better things to do than stand around documenting when each employee arrives every day. They get paid for the hours they actually work, and if they are habitually late you take that up with the employee individually. You should have a written attendance policy and enforce it. That means employees who don't follow it are disciplined or fired.
As far as the written attendance policy goes, it should also specify the procedure for taking personal time, and I'll bet that what you wrote above would be in violation of a reasonable policy. In no way would I allow my employees to use personal time as an excuse to show up late whenever they feel like it.
You're making this problem worse with a stupid policy. In addition, you're avoiding handling this directly and effectively with those who are the offenders.
If these are salaried employees, that's a whole different ball of wax.