While watching a lecture on Eulerian path, I found the following slide
Now my question is how is the above walk a path? The vertex w is getting repeated twice and then $ue_1ve_2we_3xe_4ye_5w$ should be just a walk and not a path. Can anyone please help me with my erroneous concept?
Yes, by the ordinary definitions a "Eulerian path" is usually not a "path". (Some sources might define "path" differently.) But "Eulerian path" is a common traditional term. Even though the Wikipedia article uses mostly "Eulerian trail", the actual page title is "Eulerian path".