Differences between third party and suppliers

There is no important difference, because a supplier is a third party. To identify interested parties, please read this article “How to identify interested parties according to ISO 27001 and ISO 22301”: https://advisera.com/27001academy/knowledgebase/how-to-identify-interested-parties-according-to-iso-27001-and-iso-22301//

Regarding to third party agreement vs. Supplier relationships, yes, is a case of terminology, because you can establish relationships with suppliers, and they can be considered as an interested party. Anyway, I recommend you this article about suppliers "6-step process for handling supplier security according to ISO 27001" : https://advisera.com/27001academy/blog/2014/06/30/6-step-process-for-handling-supplier-security-according-to-iso-27001/