1. The two models, definitionally
A brief recap; the definitional pillar is Why Amazon 1P, which goes deeper on the mechanics.
Third-party (3P) is the open marketplace. Brands sell their own products through Seller Central. The brand is the seller of record. The brand owns inventory, sets pricing, manages or outsources fulfillment (FBA, FBM, or Seller-Fulfilled Prime). The brand pays Amazon referral fees on each sale and receives disbursement on Amazon's 14-day cycle.
First-party (1P) is the invitation-only wholesale relationship. Brands wholesale products to Amazon. Amazon takes title at wholesale, becomes the seller of record (Sold by Amazon.ca), sets the retail price algorithmically, and handles fulfillment and customer service end-to-end. The brand invoices Amazon (or, in VendorSprout's Canadian program, invoices us as the vendor of record) on standard wholesale terms — Net 45 with favourable terms.
Both models can — and frequently do — coexist on the same brand. Different SKUs can sit in different channels. The same ASIN can transition between channels (with rankings, reviews, and A+ Content preserved). The two models are not mutually exclusive.