My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?
Of course this depends on your definition of fractal. There are nowhere-differentiable functions with graph of Hausdorff dimension 1.
Is the WM function the easiest example of a nowhere differentialbe continuous function?
For example, a nowhere-differentiable function due to Kießwetter was designed to be used with high-school students in Germany. English translation in my book: Classics on Fractals
Are these definitions equivalent
No, the definition with self-similarity is not equivalent to Hausdorff dimension > topological dimension. [Using self-similarity as a definition of fractal should be considered something to use for non-mathematicians who are curious about the subject, but have no hope to understand measures and such for the real definition.]
are the precise definition still under debate
Mandelbrot gave the definition: Hausdorff dimension strictly greater than topological dimension. He later wrote that he regretted this, and instead it should be left undefined. Others have provided other definitions. For actual mathematical papers, the authors of course state what they are proving in real mathematical language, not using the word fractal or just using it for the vague explanatory part of the paper.
Kiesswetter function, two figures from Classics on Fractals