A packet of CM points on $Y=\mathbf{SL}_2(\mathbb{Z}) \backslash \mathbb{H}=\mathbf{SL}_2(\mathbb{Z}) \backslash \mathbf{SL}_2(\mathbb{R}) / \mathbf{SO}_2(\mathbb{R}) $ is a very old notion. To each CM point corresponds a class of fractional ideals in $\operatorname{Pic}(\Lambda)$ where $\Lambda$ is an order in some imaginary quadratic field $E/\mathbb{Q}$. A "packet" of CM points in this special case is the collection of all CM points in $Y$ corresponding to a class of a fixed order $\Lambda \subset E$. Hence each order in an imaginary field defines a unique packet of CM points which carries an obvious action of $\operatorname{Pic}(\Lambda)$.
Notice that a point on $Y$ is nothing more then an orbit of the real compact torus $\mathbf{SO}_2(\mathbb{R})$ on $\widetilde{Y}=\mathbf{SL}_2(\mathbb{Z}) \backslash \mathbf{SL}_2(\mathbb{R})$. There is an analogues notion of packets for orbits of the real split torus $A=\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1}\end{pmatrix}$ on $\widetilde{Y}$. These are exactly the periodic geodesics which are parametrized by Picard groups of orders in real quadratic extensions of $\mathbb{Q}$.
The name "packet" has been introduced by Einsiedler, Lindenstrauss, Michel and Venkatesh https://arxiv.org/abs/0708.1113, https://arxiv.org/abs/math/0607815 . Why give a new name to such a well-known object? This is because it has useful generalizations to quotients of higher rank linear reductive groups $\mathbf{G}$. The cases $\mathbf{SL}_n(\mathbb{Z})\backslash\mathbf{SL}_n(\mathbb{R})$, i.e. $\mathbf{G}=\mathbf{SL}_n$, and $\mathbf{G}=\mathbf{SL}_2\times \mathbf{SL}_2$ have been studied by several people.
The idea of Einsiedler, Lindenstrauss, Michel and Venkatesh is to put under a single framework several different objects which have been studied under several names. What they have noticed that there are many cases when one has a finite collection of arithmetic significance of orbits of a torus $H<\mathbf{G}(\mathbb{R})$ on a quotient $\Gamma \backslash \mathbf{G}(\mathbb{R})$ where $\Gamma$ is a congruence lattice. These come from a single "torus orbit" on an adelic quotient. Specifically, there is a compact-open subgroup $K_f<\mathbf{G}(\mathbb{A}_f)$ and an open embedding
$$ \Gamma \backslash \mathbf{G} \hookrightarrow \mathbf{G}(\mathbb{Q}) \backslash \mathbf{G}(\mathbb{A}) / K_f$$
This embedding is a homeomorphism exactly under the "class number 1" assumption which simplifies the notation. Otherwise, the right hand side above is a finite unite of quotients of the form $\Gamma_\delta \backslash \mathbf{G}(\mathbb{R})$ for varying congruence lattices $\Gamma_\delta$. Now if we fix an algebraic torus $\mathbf{T}<\mathbf{G}$ defined and anisotropic over $\mathbb{Q}$ then we can consider the closed set
$$\mathcal{H}=[\mathbf{T}(\mathbb{A})g] \subset \mathbf{G}(\mathbb{Q}) \backslash \mathbf{G}(\mathbb{A})$$
where $g\in \mathbf{G}(\mathbb{A})$ is fixed. This is a single orbit of the locally compact abelian group $H_\mathbb{A}=g^{-1} \mathbf{T}(\mathbb{A}) g$ and carries a unique $H_\mathbb{A}$-invariant probability measure (the Haar measure which is finite due to the assumption that $\mathbf{T}$ is anisotropic over $\mathbb{Q}$).
If we project $\mathcal{H}$ to $\mathbf{G}(\mathbb{Q}) \backslash \mathbf{G}(\mathbb{A})/K_f$ (and maybe restrict to the component $\Gamma \backslash \mathbf{G}(\mathbb{R})$) we get a "packet" which is a finite collection of orbits of the real torus $H_\mathbb{R}=g_\infty^{-1} \mathbf{T}(\mathbb{R}) g_\infty$ where $g_\infty\in\mathbf{G}(\mathbb{R})$ is the archimedean part of $g$. The crux is that this finite collection of orbits in the real quotient is actually a single orbit in the larger adelic quotient. The adelic object $[\mathbf{T}(\mathbb{A})g]$ was invariant under a much bigger group $H_\mathbb{A}$. There is a trace of this action in the real setting as well which is a transitive action of a finite abelian group on the set of $H_\mathbb{R}$-orbits in the packet through Hecke correspondences. This generalizes the $\operatorname{Pic}(\Lambda)$ action for CM points.
More important then the packet itself is the $H_\mathbb{R}$-invariant probability measure it supports which is the push-forward of the $H_\mathbb{A}$-invariant measure on $[\mathbf{T}(\mathbb{A})g]$ to the real quotient. In the very special case of the modular curve this can be though of as an average over finitely many delta measures on $Y$ which is an $\mathbf{SO}_2(\mathbb{R})$-invariant measure on $\widetilde{Y}$. In general, this measure is a finite uniform average of the normalized $H_\mathbb{R}$-invariant measures on the periodic orbits in the packet.
Moreover, these collection of torus orbits in the general setting are not divisors in the algebraic sense for several reasons. The first one is that locally symmetric spaces are in general not algebraic varieties.
Lastly, in regard to the status of the mixing conjecture of Michel and Venkatesh, it seems that my paper you have mentioned and the results of Ellenberg, Michel and Venkatesh https://arxiv.org/abs/1001.0897 which apply in a restricted regime are all that we know. I would also like to stress that in the setting of the mixing conjecture there is no real added generality by using the fancy adelic construction then just $\operatorname{Pic}(\Lambda)$ orbits on pairs of CM points (with the diagonal action). The usefulness of the adelic viewpoint is important in the proof as it clarifies the role of the torus action and how to apply measure rigidity tools.
