Responding to the criticism, Evelyn told Independent.ie she can "understand his point".
She added, "I think what's he referring to is really is that they were storm winds but there was no circulation. Traditionally a storm has circulation, like a vortex, a whirling mass - like a hurricane.
"We issued a storm warning because there were storm gradient winds - but there was no circulation with it."
Ireland's Met Eireann and the UK's Met Office have a joint storm naming system. They revealed their Winter Storm names for 2017-2018 in September.
Dutton questioned how Ireland and the UK can work together on naming storms when they use "different criteria" to assess weather systems.
"This is the problem with the storm naming system," he wrote.
"The Irish Met Office, @MetEireann, have different (lower) criteria that are numerically driven, compared to the @metoffice’s impact-based criteria. How can something be a joint initiative when each use different criteria?!"
Evelyn said, "We have a joint storm naming system with the UK but [Storm Fionn] didn't affect the UK. Sometimes if a storm hits Ireland it moves on to the UK but this didn't really affect them as such.
"It doesn't matter what we call it, the point is if our system moves on to the UK they would use the name and vice versa so it doesn't really matter what the criteria are."
She added, "That presenter was saying 'this is ridiculous, Fionn isn't really a storm' but it was a storm in the sense that it fulfilled our Orange level weather warning."
In Ireland, the criteria for a storm being named are:
A wind storm with potential for significant land-based impact has been forecasted.
A severe wind event giving rise to Status Orange or Status Red weather warnings.
Consideration will also be given to rain and snow events.
Dutton went on to claim that Met Eireann has not used storm naming "sensibly".
"Storm naming is a useful tool, providing it is used sensibly," he wrote on Twitter.
"[On Tuesday], it wasn’t. 80mph wind gusts on a very exposed western coast aren’t representative of what nearly all of Ireland experienced. Just like it would be silly naming a storm because of an 80mph gust on a mountain."
Evelyn says his point is "well made" but reiterated the criteria for naming a storm in Ireland.
"His point is well made. I'm fine with it," she says. "There was no real storm circulation as such. We issue storm warnings because there were, and have been, storm force winds in the South West, very severe winds, and coastal flooding with very high waves.
"I can understand his point - he wants a storm circulation - but it fulfilled our criteria for a Status Orange warning."
On Tuesday Met Eireann issued a Status Orange and three Status Yellow warnings across the country while gusts of 137km were reported. There was also a snow-ice warning issued and a high flood risk along the Atlantic coast.