Should planar Euclidean graphs be planar straight-line graphs?
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.