Definition of likelihood ratio test Do I understand this correctly:
You fit a model with unknown parameters to a dataset. You choose the parameters so the likelihood of the dataset under the model is maximal. Let this be $L_{max, model1}$. You do the same with a more advanced model. Again you pick the coefficients of the model so that the likelihood of the dataset under the model is maximal. Let this be $L_{max, model2}$
To see which model is the best, you compare the two (maximum) likelihoods. Now what confuses me is the definition of the likelihood ratio test: 

"The likelihood ratio is the ratio of the likelihoods of the two
  models... ".

Shouldn't that be "The likelihood ratio is the ratio of the maximum likelihoods of the two models..."?
 A: You're right in the context of "submodel" testing, the likelihood ratio statistic is the ratio of the maximum likelihoods (not the maximum likelihood estimates: the maximal values of the likelihoods). 
Consider a statistical model with likelihood $l(\theta \mid y)$ where $y$ is the vector of observations generated from a distribution with parameter $\theta$ belonging to some space $\Theta$. Let $\Theta_0 \subset \Theta$ and imagine you are interested in testing $H_0\colon\{\theta \in \Theta_0\}$. 
The likelihood ratio statistic is 
$$lr(y) = \frac{\sup_{\theta \in \Theta}l(\theta \mid y)}{\sup_{\theta \in \Theta_0}l(\theta \mid y)}. $$
But when the test hypotheses are $H_0 \colon\{\theta =\theta_0\}$ vs  $H_1 \colon\{\theta =\theta_1\}$, as in the classical Neymann-Pearson lemma, then the likelihood ratio statistic is the ratio of the likelihoods:
$$lr(y) = \frac{l(\theta_1 \mid y)}{l(\theta_0 \mid y)}.$$
A: The distinction between the likelihood ratio for completely specified probability mass or density functions (simple hypotheses) & the likelihood ratio for incompletely specified ones (composite hypotheses) is sometimes expressed by calling the latter a generalized likelihood ratio. So your quote could be giving a precise definition of the likelihood ratio proper, or an vague definition (because it doesn't specify how particular parameter values are picked to calculate the likelihood for each model) of the generalized likelihood ratio.
