Reddit mentions of Finiteness: Theoretical and Empirical Foundations

So Salpfish is right when he says there's no such monolithic thing as "the infinitive." It's a term that spans a lot of different properties across languages, but the thing that unites them, as salpfish notes, is that they are "non-finite". That being said, the definition of "finiteness" is pretty tricky (to see that this is true, just look at the entire book that's been written just about the concept alone). Traditionally, a finite verb is one that shows some obligatory categories of inflection in declarative clauses. In Indo-European languages, these are tense and agreement (person/number). But this might differ from language to language.

I think a better way and maybe more productive way of thinking about things for conlanging is using the concept of clause type. A clause type is a specific kind of a clause that shares properties with other clauses of that kind. A list of some clause types to think about when you're making a language:

• Root declaratives: John kicked the ball.
  
• Root yes/no questions: Did John kick the ball?
  
• Root wh-questions: What did John kick?
  
• (Finite) Complement clause: I said that John kicked the ball.
  
• Embedded yes/no question: I asked if John kicked the ball.
  
• Embedded wh-question: I asked what John kicked.
  
• Relative clause: I saw the ball that John kicked
  
• (Non-finite) complement clause: John wants to go kick the ball
  
• Adverbial clauses (various types): I screamed when John kicked the ball
  
  There are others and this inventory will differ from language to language to language. But once you start thinking about it in this way, and I think you're getting there already, it becomes more thinking about how your conlang will accomplish various grammatical tasks rather than trying to fit it to labels like "do I need an infinitive?" You may find that you want to call the way that your conlang does one of those above things "an infinitive" but you don't necessarily have to.