I know some credit cards also have a 13.56 chip in them.
No most credit cards do not have a contactless chip in them.
When you bring a card near it, an led lights showing it senses a card.
Not quite that is showing that it has read the public ID of the card.
When I tried to see if it would light the led, it didn't
I would not expect it to.
Again, I wanted to see if my entry system could be activated by a random RFID card,
No the MIFARE cards have unique IDs on every one. Your system has to look at the number the reader returns not just the fact that the reader sees a MIFARE card.