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A Graph is Eulerian iff $\exists$ an Eulerian Cycle or all the vertices of Graph have even degree.
What is an Euler graph?
Wiki has a definition for the Eulerian graph but not for the Euler graph.

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The distinction is given at Wolfram.

The Euler graph is a graph in which all vertices have an even degree. This graph can be disconnected also.

The Eulerian graph is a graph in which there exists an Eulerian cycle. Equivalently, the graph must be connected and every vertex has an even degree.

In other words, all Eulerian graphs are Euler graphs but not vice-versa.

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