Is this argument an example of equivocation or other fallacies?

I was recently debating a libertarian and he made an argument that I believe was bad reasoning. I will attempt to break his argument down structurally. He was attempting to expose some hypocrisy or inconsistency in my argument about advocating universal healthcare as a someone who opposes nationalism.
His argument was as follows:

P1: You say oppose nationalism but want to nationalize certain industries.
P2: You cannot have the latter without the former since both rely on idea of the betterment of a nation's populace.
C: Therefore, you support nationalism as well.

Are there more than one fallacy in this? I believe he is equivocating with the word 'national'

One of the first rules of debate (or effective communication) is make sure all parties have a common understanding of the terms used. "Nationalism," as commonly defined means: identification with one's own nation and support for its interests, especially to the exclusion or detriment of the interests of other nations. Notice that technically, the definition does not require the latter part (detriment of other nations). So while this may be what you object to most you need to make this clear.

Now let's look at universal healthcare. Does this mean "identification with one's own nation and support for its interests"? I would think so. Few people would have a problem with this part. But what about "to the exclusion or detriment of the interests of other nations"? Here is where knowledge of economics and politics are needed. Would such a program take resources away from other programs that support people from other nations? I don't know, but the answer is important because if so, it technically would seem to be nationalistic.

Here is where equivocation can be argued. "Nationalism" also generally refers to a philosophy rather than a specific policy. This is the equivalent of someone being labeled a "socialist" because they support public education. So you might qualify your remarks about nationalism by saying something such as "I reject nationalism as a general philosophy but that doesn't mean that I reject all nationalistic policies" and make the point that the two are not the same (and feel free to use my socialism example as an analogy).

Dr. Bennett's analysis is correct and I have nothing to add to it.

What we've got here is failure to communicate.

If not a Strawman conflation of nationalism v socialism.

P2: You cannot have the latter without the former since both rely on idea of the betterment of a nation's populace.
C: Therefore, you support nationalism as well.

This is clearly a hasty generalization, and or Ambiguity Fallacy

See Dr. Bo's:

Ambiguity Fallacy
(also known as ambiguous assertion, amphiboly, amphibology, semantical ambiguity, vagueness)

Description: When an unclear phrase with multiple definitions is used within the argument; therefore, does not support the conclusion. Some will say single words count for the ambiguity fallacy, which is really a specific form of a fallacy known as equivocation.

Logical Form:

Claim X is made.

Y is concluded based on an ambiguous understanding of X.