This is a general point of confusion for me, which is why the question is going to be a bit vague. Think of the main question as: What is the right way to think about this?

I will present the question in the context I was thinking of, but it may be that the ideas are more general.

Let's say we have a morphism $X \rightarrow Y$ of relative curves over a complete DVR, $R$. Let's also assume that $Y$ is a nodal curve (meaning its closed fiber is nodal). In fact, for simplicity, assume that $Y$'s closed fiber is made up of exactly two irreducible components meeting at a node. There are many ways of blowing that "point" up: if the node is given by $xy=t^n$, we may blow up at $(x,t)$, $(x,t^2)$, and so forth.

Now, it may be that $X$ is not nodal. What I would like to compare is blowing up at all the pre-images of the node in $X$, and blowing up at the node in $X$ and then normalizing.