To what extent the statement "Data is normally distributed when mode, mean and median scores are all equal" is correct? I read that normally distributed data have equal mode, mean and median. However in the following data set, Median and Mean are equal but there is no Mode and the data is "Normally Distributed":
$ 1, 2, 3, 4, 5 $
I am wondering to how extent the statement is correct? Is there a more accurate definition for "normal distribution"?

It is not correct at all.  Any unimodal probability distribution symmetric about the mode (for which the mean exists) will have mode, mean and median all equal.
For the definition of normal distribution, see e.g. Wikipedia.
Strictly speaking, data can't be normally distributed, but it can be a
sample from a normal distribution.  In a sample of $3$ or more points from a continuous distribution such as the normal distribution, with probability $1$ the data points will all be distinct (so there is no mode), and the mean will not be exactly the same as the median.  It is only the probability distribution the data is taken from that can have mode, mean and median equal.
