Easy going advisor versus expert advisor and research experience
How you approach the professor for acceptance would be very similar to how you speak to your current advisor (should you choose to*) about your decision:


Be objective, there is no need to mention anything about either academic in a personal way, this decision is purely one for an academic reason - you want to be 'closer' to where current active research is happening and amongst those actively involved in research in your field.
Related, be polite and direct to the point.


The fact that the professor has expressed interest in writing a paper with you is a strong indication - perhaps accept this offer and at that stage mention that you wish to pursue your research under his guidance and to work within his labs.

Once it is definite that you'd get in and although you are not obligated to explain to your current advisor, it is advisable to explain clearly and positively about the opportunity to work directly with one of the top researchers in the field - I am sure your advisor would understand.