I don't know if I can state a clear single "definition", but hopefully I will be able to sort out some of the concepts and the confusion.

integrability is sometimes associated with having a closed form solution

This, I think, is categorically not true. At least in the usual sense of 'closed form'. If you take the Lieb-Liniger model, which is I believe one of the seminal examples of an integrable system, the solution obtained is in the form of a set of integral equations, that the authors proceed to solve numerically. This is not 'closed form'.

integrability is sometimes associated with having infinitely many conserved quantities

This is the definition I am familiar with, but it requires caution and there are some subtleties. Namely, every system at the thermodynamic limit has an infinite number of conserved quantities: the projectors onto the eigenstates of the Hamiltonian $|\psi_n\rangle \langle \psi_n |$. Therefore, this definition alone is not enough. One needs an infinite number of conserved quantities that are 'not trivial' in some sense. Sometimes they are defined by being with local support, but I am not sure that this is enough or unique. However, it usually guaranteed that if one has a solution of the system in terms of the 2-particle scattering matrix and the associated Yang-Baxter equation, one can construct this infinite number of conserved quantities.

integrability is sometimes kind of like the opposite of chaos

 

integrability is sometimes kind of like the opposite of thermalization

These two are related, as I understand them, and the notion is generally derived from the existence of the infinite number of conserved quantities. The idea is that if we have an infinite number of 'non trivial' conserved quantities, then we can describe the macroscopic observables using them, and then the observables keep their value throughout the time-evolution. This, of course, contradicts thermalization and chaos, in the sense that if a system is prepared in some state it will keep its initial observables, instead of thermalizing. However, this is a subject of a very lively debate, surrounding the questions of what is exactly the nature of the the conserved quantities, whether or not the 'eigenstate thermalization hypothesis' is true or not, and how can one generalize integrability to 'quasi-integrable' models.

I think that as in many other topics in contemporary physics, there is no clear definition of integrability. Once it was related to a system having an exact solution (usually via the Bethe-ansatz method or one of its relatives), and the infinite number of conserved quantities was a feature / definition depending on your point of view. Nowadays the term migrated and expanded, together with the interests of the community.

I don't know if I can state a clear single "definition", but hopefully I will be able to sort out some of the concepts and the confusion.

integrability is sometimes associated with having a closed form solution

This, I think, is categorically not true. At least in the usual sense of 'closed form'. If you take the Lieb-Liniger model, which is I believe one of the seminal examples of an integrable system, the solution obtained is in the form of a set of integral equations, that the authors proceed to solve numerically. This is not 'closed form'.

integrability is sometimes associated with having infinitely many conserved quantities

This is the definition I am familiar with, but it requires caution and there are some subtleties. Namely, every system at the thermodynamic limit has an infinite number of conserved quantities: the projectors onto the eigenstates of the Hamiltonian $|\psi_n\rangle \langle \psi_n |$. Therefore, this definition alone is not enough. One needs an infinite number of conserved quantities that are 'not trivial' in some sense. Sometimes they are defined by being with local support, but I am not sure that this is enough or unique. However, it usually guaranteed that if one has a solution of the system in terms of the 2-particle scattering matrix and the associated Yang-Baxter equation, one can construct this infinite number of conserved quantities.

integrability is sometimes kind of like the opposite of chaos

integrability is sometimes kind of like the opposite of thermalization

These two are related, as I understand them, and the notion is generally derived from the existence of the infinite number of conserved quantities. The idea is that if we have an infinite number of 'non trivial' conserved quantities, then we can describe the macroscopic observables using them, and then the observables keep their value throughout the time-evolution. This, of course, contradicts thermalization and chaos, in the sense that if a system is prepared in some state it will keep its initial observables, instead of thermalizing. However, this is a subject of a very lively debate, surrounding the questions of what is exactly the nature of the the conserved quantities, whether or not the 'eigenstate thermalization hypothesis' is true or not, and how can one generalize integrability to 'quasi-integrable' models.

I think that as in many other topics in contemporary physics, there is no clear definition of integrability. Once it was related to a system having an exact solution (usually via the Bethe-ansatz method or one of its relatives), and the infinite number of conserved quantities was a feature / definition depending on your point of view. Nowadays the term migrated and expanded, together with the interests of the community.