Can I successfully adapt to an advisor who will not push me to work?
To be sure, chooosing your advisor is one of the most important decisions in your academic career.  That is not to say that people don't switch advisors for one or more reasons (they do), but it's probably best to take the time to figure it out now (which it sounds like you are doing by asking the question in the first place!) so you aren't faced with that decision later.

It seems to me that you may be asking the wrong question, to some extent.  Relying on your advisor to push you is more or less the opposite of a good way of growing into being an independent researcher -- who is going to push you when you become that assistant professor and you're on your own?  Furthermore, the last thing you want your advisor writing in his/her recommendation letter is, "Great researcher when pushed to complete the work!"  You also don't want your advisor to think that he/she has to push you -- this may just lead to a poor working relationship between the two of you.

My advice is to ask Professor Nice Guy to be your advisor, but also start working now towards limiting the procrastination on your own.  That advice is predicated on your comment that your research matches his the best; that should be the driving factor and not some grand idea that you'll get pushed harder by one prospective advisor or another.  Tips for how to get a good plan together are outside the scope of this topic, but at the very least you could start by fixing a date you'd like to graduate and working back from there.

As for meeting with the other professor regularly, by all means do that if he is amenable to the idea.  I met regularly with my advisor and another professor and all parties were happy with the arrangement.