Confusion: different definitions of MAP estimation in Graphical Models (MRFs) 
*

*The "classical" MAP estimation: $$\hat\theta = \arg\max_{\theta}P(\theta|\mathbf{x})$$ where $\mathbf{x}$ are the observations and $\theta$ are the parameters. 

*In this book chapter (page 6, second item), MAP estimation for a MRF is to maximize $P(\mathbf{x}|\mathbf{z},\theta)$ w.r.t $\mathbf{x}$, where $\mathbf{x}$ is the sequence of states, $\mathbf{z}$ is the set of observed data.

*In this paper (page 2), that is to maximize $P(\mathbf{x}|\theta)$.    
I would appreciate it if somebody could make the connection among these three clearer. Thanks in advance.
 A: You confusion shows that you are a very exact person!  ;-) 
Assumptions on the notation: 
 - $\theta $: Parameters 
 - $ \mathbf{x} , \mathbf{z} $: Variables 
Among Bayesian people, when someone talks about estimation, they refer to estimation of almost anything. See this: 
ftp://ftp.cs.utoronto.ca/pub/radford/bayes-tut.pdf
In Neal's tutorial, page 4 he estimates the parameters of the model, given input data, using the probability of the posterior. Finding the maximizer will give you the most-probable estimates (your 1st definition).
\n 
See, your 2nd and 3rd definition are basically the same. In some applications you might have some input $z$, which is usually known. Given the parameter $\theta$, the probability of seeing $X = x$ is: 
$$
p(X = x | Z = z , \Theta = \theta)
$$
The most probable observation is (you second defintion) :
$$
\arg\max_{x} p(X = x | Z = z , \Theta = \theta)
$$
\n 
In a model, assume you don't have any input (like mixture of gaussians model). Given the parameter $\theta$, the probability of seeing $X = x$ is: 
$$
p(X = x | \Theta = \theta)
$$
The most probable observation is (your third defintion) :
$$
\arg\max_{x} p(X = x | \Theta = \theta)
$$
