Should I write about quitting a currently Masters program in a Motivation Letter?
In my opinion, from a US perspective, you should definitely mention your current Masters program and reasons for wanting to shift to a research-based degree.  

The language you use in your question, and the brief length you allot to your current program and reasons for deciding on different path, are fact-based, honest, and positive.  

The questions I have are:


Do you have interim grades from your current program that you can include in the letter?
Do you have a current professor who would be willing to discuss your talents and performance, and your promise as a student in a more research-based degree?


And finally,


How much time is left to finish your current program?  If it is a year or less, I suggest you just finish it. The process of applying and starting a new program will likely take most of a year anyway, and this way you will have the degree on your CV and your reasons for pursuing another Masters as a path to a PhD still hold.


I don't know about international universities, but in the US, most science and engineering schools award a Masters on the way to a PhD, it's pretty seamless.

Good luck with your studies either way, and on your future.