Taxpayer
In short
The business that owes the tax, not the customer who pays the price. On a contracting project, which business that is depends on the classification.
In full
Transaction privilege tax is levied on persons on account of their business activities. The taxpayer is the business. A customer may bear the cost inside the price and never becomes the taxpayer by doing so.
On a Modification project the taxpayer is the prime contractor, on the gross receipts of the contract. A licensed subcontractor holding a Form 5005 is not liable, which is not the same as not being required to report — it still reports its gross receipts and deducts them.
On an MRRA project the answer depends on how the materials were bought. Where the contractor paid tax to the vendor at purchase, the vendor is the taxpayer on that sale and the contractor is a purchaser. Where the contractor bought exempt, the contractor becomes liable for an amount equal to retail transaction privilege tax and is the taxpayer on it. A contractor performing only MRRA work who always pays at the register is not a transaction privilege taxpayer at all, which is why that contractor generally needs no TPT license.
The term is not defined once for all of Arizona tax law. Each code defines it where it needs to, and the Model City Tax Code carries its own definition, referenced by the city construction contracting section. The definition belonging to the provision in hand governs; one meaning should not be assumed across the chapter.
A.R.S. §§ 42-5008, 42-5008.01; ADOR Contracting FAQs; Model City Tax Code § 415(b)(10)