I am leaving current PhD position to move to X university. Should I inform X university that I am currently a PhD student?
I don't see any reason not to just lay it out as you have here. "I think that the program here would better suit my needs and I think I'll get better research and career advice here." 

People understand that other people want to change and move "up" when possible. 

But I would also take some care that you fulfill your promise to your current advisor about finishing the research to the maximum level possible. You say it is a short-term commitment, so I think it wouldn't be overly burdensome to continue until it is done or until circumstances make it impossible to continue. In case you have to leave it, try to do what you can to leave it in a stable state in which others can, if necessary carry it on. That way your current advisor is more likely to remain a supporter in your future. 

But your proposal and your actions seem perfectly normal and natural to me.