Johansen test explanation
I think that the wikipedia steps are not completely correct. When you perform Johansen cointegration test you first have to pretest the data to find if they have the same order of integration. That is all variables have to be either $I(1)$ or $I(2)$ etc. (see Verbeek, 2008). There are some cointegration tests and models that relax this assumption but Johansen is not one of them. But otherwise the steps 1-3 are correct. 

$\Phi D_t$ are indeed seasonal dummies. However, note they are actually not standard part of the model, they are added when you think series can be affected by seasonal trends and if you dont use some other seasonal adjustment technique or if you dont already have data provided with seasonal adjustment from the source.  

$\mu$ is the constant or 'drift' term. It allows your regression to have different intercept than zero. 

You are correct $\Pi$ are matrices of coefficients but dont forget about $\Phi$. Any coefficient you add there has to be estimated - I know its just nitpicking but still.

The co-integration vectors come from the $\Pi$ in the second equation. This matrix can be decomposed as $\Pi = \gamma \beta'$ where $\gamma$ is the matrix of weights with which the cointegration vectors enter the model and $\beta$ is actually the matrix of cointegrated relationships. I am not really exactly sure what you mean by choosing cointegrated vector from this matrix. Its not like you just somehow extract one of the vectors somewhere into some separate vector.

Afterwards the Johansen test basically tests the rank or the number of columns of the cointegrated matrix (With dimensions $k$x$r$ you test for # of $r$). Intuitively this works because you need one column to represent each cointegrated relationship (note there might be multiple cointegrated relationship at the same time).

The rank of the matrix is tested either by trace or eigenvalue test as you correctly mentioned. The null of trace test is $H_0: r \leq r_0$ against $H_a: r &lt;r_0 \leq k$  and the max eigenvalue test has the same null but alternative hypothesis is slightly different $H_a: r = r_0+1$. However, this might be hard to read in plain English. In practice if you would have only two variables both tests would boil down to first testing 0 cointegrated ($r=0$) relationship vs 1 $(r=1)$, second less or equal to 1 cointegrated relationships $(r\leq 1$) vs 2 ($r=2$), third less or equal to 2 coint. rel. $r\leq 2$ vs 3 ($r=3$). You dont really pick from these vectors although the number of cointegrated vectors can affect functional form of VECM, but with just two variables you are looking for $r=1$, if your test finds there should be more you are missing some important variables in your model.

Also yes $\Delta$ is the first difference $\Delta y_t = y_t - y_{t-1}$. Also including first differences assumes that the variables are all $I(1)$ if they are $I(2)$ you need at least second differences. The reason for that is that outside the cointegrated lagged variables the 'contemporaneous' variables must be stationary otherwise your ECM will be biased.  The whole justification of including level variables in ECM is that as long as there is a cointegrated relationship between those their combination will be stationary and their coefficient will be super consistent, but this does not extend to the variables that are added to the model outside the cointegrated relationship.

References:
Verbeek, M. (2008). A guide to modern econometrics. John Wiley &amp; Sons.