Does the choice of normalization change dramatically the result of a KMeans
Yes. Feature scaling can completely change the clustering result.

People usually scale data to [0:1] or to have a standard deviation of 1.

However, that is nothing but a heuristic.

In many cases, the need for scaling is nothing but a symptom, caused by inappropriate data for the method. You can't just fix this by some naive scaling, but it's just a hack that often works.

For statistically meaningful results, all axes should be scaled to reflect attribute relevance, such that a difference of 1 unit is of the same importance in each attribute.