Recently I've been doing reading in the subject of orthogonal polynomials on the real line (OPRL). Such OPs arise in solving the three-term recurrence relation $$x u_n=a_{n+1}u_{n+1}+b_{n+1}p_n+a_{n}u_{n-1}$$ with parameters $\{a_n,b_n\}_{n\geq 1}$ and a suitable boundary condition for $(u_0,u_{-1})$. What I'm seeking clarity on is the standard terminology for OPs corresponding to various choices of boundary conditions.
For instance, in Barry Simon's works (for example, in his lecture slides) he defines OPs of the 1st kind $\{p_n\}_{n\geq 0}$, with $(p_0,p_{-1})=(1,0)$, and OPs of the 2nd kind $\{q_n\}_{n\geq 0}$ with $(q_0,q_{-1})=(0,-1)$. In addition, he defines Weyl solutions $g_n(x)=m(x)p_n(x)+q_n(x)$; here $m(x)=\int (x-z)^{-1}d\mu $ is the Stieltjes transform of $d\mu$, the orthogonality measure of the first-kind OPs (i.e. $\int d\mu(x) p_n(x)p_m(x)\propto \delta_{mn}$).
By contrast, other sources in the literature (e.g. Assche's 1991 survey paper on OPs) refer to the above OPs of the second kind as 'the (first) associated OPs' and to the Weyl solutions as 'functions of the second kind.'
Now, these two naming schemes do not contradict each other; for instance, 'functions of the second kind' are not polynomials and so are distinct from OPs of the second kind. But the terminology is different, and it's not clear to me which is more appropriate nowadays. Can someone clarify which terms are more standard?