The term modified normal distribution (or equinormal distribution) may refer to any number of distributions that are similar in appearance to a normal distribution (in other words, it’s simply a normal distribution modified in some way).
For example, the t distribution is one example of a modified normal distribution .
The term can also refer to normal variance mixture distributions described by Romanowski  that modify the normal distribution to fit the variation seen in real life data sets. Romanowski called these modulated normal distributions and — as they are particular instances of normal variance mixtures — can better be described as mixture distributions.
The actual term “modified normal” is usually used in a loose sense to describe the non-normal behavior of a particular distribution, rather than to describe a specific distribution. That’s because most “non normal” distributions have their own names. For example, positively skewed, unimodal distributions might better fit a gamma distribution or lognormal distribution.
Other types of modified normal distribution
A zero-modified normal distribution is a normal distribution modified to put extra probability mass at 0. In other words, it’s a kind of mixture distribution where part of the population comes from a normal distribution but the rest of the population is all zeros .
An exponentially modified normal distribution is a three-parameter distribution that is a generalization of the normal distribution for skewed cases.
 Romanowski, M., Green, E. Tabulation of the modified normal distribution functions. Bull. Geodesique 78, 369–377 (1965).
[3[ Environmental Statistics with S-PLUS p.192