I think to answer this it is best to first go over definition of identification.
Following Stachurski (2016) identification or identifiability (I omit formal description since its also in that wikipedia article you provided in your comment):
means that the parameter vector associated with unknown distribution can eventually be distinguised from the data.
Hence identification is more or less what people usually call estimation. For example, in OLS ($y= X\beta+e$) where the $\beta$ coefficient is:
$$\hat{\beta} = (X' X)^{-1}X'y$$
it can be proven that $\hat{\beta}$ can only be identified when the $X'X$ matrix is invertible otherwise $(X'X)^{-1}$ is not defined and you simply wont be able to calculate the $\beta$ or R or Python or Stata would give you error message, like for example where you have perfect multicolinearity.
Every model you can think of has some identification conditions - hence its not really appropriate to talk about types of identification, identification means the model can estimate the parameters and every model has its own conditions for identification of the parameters.
Consequently the process of estimating any coefficients is usually called identification strategy. If by asking about different types of identification you mean different models/ identificaiton strategies, a good-all-purpose textbook is Pesaran (2015). Verbeek (2008) is good intermediate introduction textbook, if you look only for time series analysis Hamilton (1994) is classic although bit dated now. For treatment evaluation I recommend Angrist & Pischke (2008). I wont go over all possible types of models as nowadays you have such a variety its impossible to make exhaustive list in answer on SE.
Next when we talk about weak or strong identification we mean by weak identification that the estimators and test statistics are not well approximated by their standard asymptotic limits because of limited information in the data. Basically, the point is that identification in itself does not guarantee the coefficients are consistent and unbiased only that they can be estimated and usually the term weak identification is applied when the aforementioned are not guaranteed and strong identification when they are. Also I mostly seen these terms being applied to the IV, GMM or other instrument using estimator where weak identification is often used as a synonym for the fact that the first stage or instruments are weak. But I dont see people denoting weak or strong as a type so I dont know if this is what you mean by that. To see when the coefficient estimates are consistent and unbiased you can again for each separate model see in the literature I recommended above.
References:
Angrist, J. D., & Pischke, J. S. (2008). Mostly harmless econometrics: An empiricist's companion. Princeton university press.
Hamilton, J. D. (1994). Time series analysis (Vol. 2, pp. 690-696). New Jersey: Princeton.
Pesaran, M. H. (2015). Time series and panel data econometrics. Oxford University Press.
Stachurski, J. (2016). A primer in econometric theory. Mit Press.
Verbeek, M. (2008). A guide to modern econometrics. John Wiley & Sons.
