I think you're confused about the definition of polyandry. Firstly, polyandry normally refers to sustained relationships, not simply mating. In birds this means nest biulding, incubating eggs, and feeding young. If a pair of birds raise chicks in one nest, and neither interacts with another nest of chicks, then they are a monogamous pair - it doesn't matter if there are other chicks with their genetic material.
Also, polyandry normally refers to females taking more than one male partner, while males stick with that single female. The article states that most females "cheat" on their partners but a similar percentage of males do the same.
The dunnocks given as an example often form trios where 2 males and a female collectively raise chicks, with the young having a mixture of paternities. The jacanas have a system where males build nests and females claim up to 4 males, and defends all her nests while the males raise the young. These are very different to the standard bird system, where a female bird may have many sexual partners, but only cares for one set of young.