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An Euclidean graph, by definition is

A weighted graph in which the weights are equal to the Euclidean lengths of the edges in a specified embedding

and a graph is called planar if

it can be drawn in a plane without graph edges crossing

lastly, a planar straight-line graph (PSLG) is

embedding of a planar graph in the plane such that its edges are mapped into straight line segments.

By these three definitions, I cannot conclude that if an Euclidean planar graph should be a PSLG or not. For instance, given an Euclidean non-planar graph:

enter image description here

I can convert this graph into an Euclidean planar graph

enter image description here

by sticking with the definitions. Given the definitions above, the former one is a non planar straght-line graph whereas the latter one is a planar graph. Assume that the length of each edge is preserved. Then is the latter one still an Euclidean graph?

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Fáry's theorem states that every planar graph can be drawn in such a way that its edges are (non-crossing) straight lines. Hence every planar graph is a planar straight-line graph.

However, this doesn't really have any bearing on the definition of Euclidean graph. According to the definition you link too, the edges need not be straight. A Euclidean graph is a planar graph in which the edge weights are their lengths in some specific (planar) embedding. In particular, a non-weighted graph isn't Euclidean any more than a cucumber.

Finally, the first graph in your question is planar – it's just that your specific embedding isn't a planar embedding.

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