How to apply for a new PhD while being enrolled in one? What to provide as motivation letter, CV and references?
First, your situation isn't as rare as you might think. It is also moderately common to want to change programs, or at least advisors. Changing because you aren't getting the supervision that you want and need is also not that unusual. 

It may be a bit less usual, however, if you want to change, but don't want to change "what you do" as well. 

I would prepare an application package, including CV and SOP just as if you weren't already in a doctoral program. What have you done, what are your goals. Then, modify the package "just enough" so that the CV shows the current program, and the SOP indicates you are looking for a "more compatible" position that "better fits your needs". This should represent a small part of your application, not the major focus. In interviews it is enough to say that your advisor "isn't sufficiently helpful." 

But I wouldn't hide the fact that you are already in a program or that you and your advisor aren't especially compatible. If it were later learned you might suffer for it. My general opinion is that you can and should be honest in all things - when possible. 

You seem to be on a search for an advisor who can help you become successful. That is a worthy goal. Focus on that and don't think of the past as a failure, but just as a difficult situation. 

And, before you choose another advisor, make sure that you get some information, somewhere, possible from him/her, about expectations on both sides. Sometimes other students are willing to share their perceptions, but you can also just explore it like any other thing with the person. If they aren't comfortable with this, they may not be the best choice. 

But expect that any new advisor that otherwise meets your needs will want to steer you in a (slightly?) different direction. In your case it might be best to agree.