Hey there @tdelillo , great question.
I would first ask them what their application does, which you have undoubtedly done.
Based on the features they provide, I would look at how those features feed information back into Canvas, and test it.
For example, if they are a test engine, and they say they integrate, then I would expect to be able to create an assignment that is of type "External Tool" that would launch their test interface. And, when students complete the test I would expect their application to push those grades back into the Canvas gradebook.
Or, if they were a plagarism app, I would expect to create an assignment typ of "External Tool", and when a student submits their a paper it gets automatically pushed through the plagarism detection. Perhaps in this case you would have access to an instructors report, or the custom assignment would display the plagarism results.
Pearson is a good example, their e-Books are capable of pushing assignments and grades directly into the Canvas.
If I had to sum it up in one sentence: LTI allows an institution to integrate multiple applications to create a seamless user experience, where the user has a single login to a single system to access all of the tools they need.
If all they have is a link that takes the student outside of the Canvas environment, then they really are not integrated with Canvas. That is not LTI.
Wikipedia gives a pretty good high level definition of LTI:
Your tool privder would be more interested in IMS Global for more technical details:
Hope that helps 
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