Q:

Alternative method to k-means that can be "guided" researcher's intuition?

I am trying to do a simple k-means clustering to my dataset. The result I get it the one that can be seen below:
However, the result I would like to have, as it corresponds to geographical areas, looks something like this:

Is there a proper way to to give weight to x axis values or limitations regarding the euclidean distance between the centroids? For example, adding some restrictions regarding the result (the euclidean distance between the two points of the same cluster should not be larger than X).
Or even a different method than k-means? My latest efforts with GMM seem to do a better job, although some 'manual' interventions (the restrictions regarding the euclidean distance) have been included.

A:

Your intuition is not objective.
You are being misled by a bad plot. Your data actually looks more like this:

Are you still convinced your "intuitive" solution is better? Split your data with a threshold on the x axis, and compute the cost function.
Just kidding. Normalize your data and this toy example will work.
If you intend to incorporate user feedback, metric learning may be worth looking into. If you learn a linear transformation, you can run k-means on the scaled data. Have a user select instances from the same and from different clusters to learn the projection.

