The Uno is a 5V system, so that wiring has 5V signals going into a 3V3 reader. As you were told before you have to cut the output signals from the Arduino down to 3V3. This can be done with a level shifter or a potential divider using a 510R and 1K resistor. Do not cut down the output of the reader, that is the MISO line.
All cards have an ID which you should be able to read with any reader.