What is the name for this form of "influence"?
I have some ideas about your first example. During my course of social psychology a learned about self/social categorization, two concepts that shape how individuals perceive groups. Here are some quick definitions.

Self categorization:


  the process of seeing oneself as a member of a social group (Smith, Mackie &amp; Claypool, 2014)


Social categorization:


  The perceptual classification of people, including the self, into categories (Forsyth, 2014).


According to self and social categorization people automatically view themselves as part of a group and others as part of a group also. I think Phil in your example still views himself as part of the group "catholics" (even though he does not know much about it) and Fred as part of the group "Evangelics". After that there are quite a view phenomenons that suggest people care about their beliefs and about their group status. One good example in this context is the ingroup-outgroup bias:


  The tendency to view the in-group, its members, and its products more positively than other groups, their members, and their products. Ingroup favoritism is more common than outgroup rejection (Forsyth, 2014).
  So, Phil might realize that he is part of the group "catholics" and subsequently becomes more defensive. 


Also it could be possible that Phil experiences some form of cognitive dissonance (an adverse psychological state that occurs when an individual simultaneously holds two conflicting cognitions (Forsyth, 2014)), because he claims he is part of the catholic group but is blatantly confronted that he barely knows anything about his group. This might lead to the attitude change you suggest, that he wants to learn more about catholicism. 

Of course this is more a description of how processes in groups can work and how people influence each other based on these processes. It is in no way a description of a scientifically backed up process/theory, although the terms used in my description are of course backed by science.

As for the second example, I personally believe there could be a lot of factors that lead to Jake's decision of joining the army. I think it would be hard to pin it down on the few words of the army veteran.

However, I am not an expert on social psychology. So I hope this information helps understanding social processes but it might not be a complete and conclusive answer. If you are interested in social psychology and group dynamics then I recommend to check out Forsyth's book Group dynamics. It explains concepts wihtin the field very well and uses informative examples. Hope this helps in some way.

References

Forsyth, D.R. (2014). Group dynamics (6th ed.). Belmont, CA: Cengage Learning. ISBN- 13 9781133956532

Smith, E.R., Mackie, D.M., &amp; Claypool, H.M. (2014). Social Psychology (4th ed.). New York, NY: Psychology Press. ISBN 978-1-84872-894-3