I think this question is a little subject to opinion, so hopefully it would be considered okay to share mine.
At that time there were a lot of different approaches for how to fit Lie groups into a broader algebraic context. The end of the 1800s saw much debate over things like this (a nice book on this by Crowe: https://www.researchgate.net/publication/244957729_A_History_of_Vector_Analysis). I would say it took at least 40 years to clean things up into a more streamlined system. And some elements of it still merit looking back at the original sources! (I've gotten some interesting results of my own by updating ideas from the early 1900s into modern methodology.)
Sometimes just comparing one person's notation and terminology with another's brings out enlightening relationships that, in hindsight, seem obvious. But carrying out those comparisons is non-trivial and often takes a long time to occur.
The first question is false as stated.
By Artin's encoding, geodesics on $SL_{2}(\mathbb{R})/SL_{2}(\mathbb{Z})$ corresponding to continued fractions, and the geodesic flow corresponds to the shift.
It's easy to find one fraction where you'll see any given prefix (hence dense), but you won't be equidistributed (say think about larger and larger blocks composed out of $1$'s).
The situation is the same even for cocompact (hyperbolic) homogeneous spaces, and relays on the fact that the corresponding dynamical system is a Bernoulli system, see for example the survey by Katok in the Clay Pisa proceedings for more information about the encoding.
In the case where the manifold is a Nilmanifold, the answer is indeed true, which follows from say Furstenberg's theorem about skew-products (when you use both the topological version and the ergodic version).
Finer (quantitative) results are probably attained by Green-Tao (see Tao's post about the Nilmanifold version of Ratner's theorem).
In the toral case, this boils done to merely Fourier series computations and Weyl's equidistribution criterion or so.
In the higher rank (semisimple) case, things get more complicated, as one might think about multi-parameter actions, and then the measure-classification theorem by Lindenstrauss kicks in, but it was observed by Furstenberg in the $60$'s (and maybe before that) that even for multi-parameter actions, there might be dense but not equidistributed orbits.
Maybe the easiest toy model to think of is to think about the multiplicative action of $<2,3>$ as a semi-group on the torus $\mathbb{R}/\mathbb{Z}$, and start at say a Liouville number for base $6$. This action is some $S$-adic analouge for a higher-rank multi-parameter diagonalizable action.
Edit - to address the revised question, here the geometrical settings are being addressed more intimately.
In the case of homogeneous spaces ($G/\Gamma$, or you can take the appropriate locally symmetric space as well), where $G$ is semi-simple say, then the geodesic flow is ergodic (it follows for example from the Howe-Moore theorem, or from the Bernoullicity theorem I've mentioned above). As a result, a simple application of the pointwise ergodic theorem will tell you that for almost every point and every direction (the approperiate measures here will be the Liouville measure on the unit tangent bundle, which is really where the geodesic flow "lives"), the orbit is equidistributed.
For the variable curvature case, as long as some natural conditions are met (say an upper bound on the sectional curvature making it negative everywhere), the dynamical picture is pretty much the same (but the proofs are significantly more involved, as you don't have rep. theory at hand).
Again in the Nilmanifold case, the situation is much more simple, the toy model for that is tori, where the question of rationality implies both density and equidistribution.
I will address the Andre-Oort question in the comments, as I'm not an expert on this subject.
Best Answer
One class of examples in a compact, connected Lie group $G$ of rank $r$ are the conjugacy classes of codimension $r$. Taking an element $a\in G$ whose centralizer is a maximal torus (a generic condition), the conjugacy class of $a$ is a submanifold $C_a\subset G$ of codimension $r$ whose normal plane at $a$ is the tangent at $a$ to $Z_a$, the centralizer of $a$ in $G$, which is a flat, totally geodesic submanifold. By the $G$-homogeneity of $C_a = G/Z_a\subset G$, the fact that this holds at $a$ implies that it holds at all point of $C_a$.
This gives an $r$-parameter family of mutually noncogruent examples.
There are other examples: When $G=\mathrm{SU}(n)$, and $a = \mathrm{diag}(\lambda_1,\lambda_2,\ldots,\lambda_n)$ is a diagonal element for which the $\lambda_i^2$ are all distinct, the submanifold $$ M_a = \{\ g_1 a g_2\ |\ g_1,g_2\in\mathrm{SO}(n)\ \}\subset\mathrm{SU}(n) $$ is homogeneous under the isometry group of $\mathrm{SU}(n)$ (endowed with its biïnvariant metric), and its tangent plane at $a$ is orthogonal to the diagonal maximal torus $T\subset\mathrm{SU}(n)$, so, by homogeneity, its tangent plane at any point is orthogonal to a flat, totally geodesic submanifold of $\mathrm{SU}(n)$. Hence it has an abelian normal bundle.
This gives another $r=n{-}1$ parameter family of mutually noncongruent examples of codimension $r$, distinct from the conjugacy classes.
Using the methods of exterior differential systems, one can show that, when $n=3$, these two families account for all of the codimension $2$ submanifolds of $\mathrm{SU}(3)$ with abelian normal bundle, in the sense that any connected submanifold $M^6\subset\mathrm{SU}(3)$ with abelian normal bundle is, up to ambient isometry, an open subset of one of the examples listed above. The argument that I have written out is a calculation using exterior differential systems and the moving frame, but, when I have time, I can sketch the proof, if there is interest.
Addendum 1: I had a flight with some time to look at the other two compact simple rank 2 groups. It turns out that all of the connected codimension two submanifolds with abelian normal bundle in $\mathrm{SO}(5)$ and $\mathrm{G}_2$ are (open subsets of) homogeneous compact ones as well. In each case, there is one additional $2$-parameter family of examples beyond the principal conjugacy classes.
Addendum 2: I also checked the rank $r=3$ case $G = \mathrm{SU}(4)$, and found that every connected codimension $3$ submanifold of $G$ with abelian normal bundle is an open subset of one of the two types of homogeneous examples listed above. Since any codimension $2$ submanifold of $\mathrm{SU}(4)$ that is foliated by codimension $3$ submanifolds of $\mathrm{SU}(4)$ with abelian normal bundle will have abelian normal bundle, it follows that there are many non-homogeneous codimension $2$ submanifolds of $\mathrm{SU}(4)$ with abelian normal bundle.
In fact, it now seems likely that, for any compact simple group $G$ of rank $r>1$, there are two $r$-parameter families of homogeneous codimension $r$ submanifolds with abelian normal bundle and every connected codimension $r$ submanifold with abelian normal bundle is, up to ambient isometry, an open subset of a homogeneous one. The two families are as follows: The first family is the family of principal conjugacy classes in $G$, and the second family is the set of principal orbits of $K\times K$ acting by left and right multiplication in $G$ where $G/K$ is a symmetric space of rank $r$. For example, when $G=\mathrm{SU}(n)$ $(n\ge3)$, $K=\mathrm{SO}(n)$; when $G = \mathrm{SO}(n)$ $(n\ge 5)$, $K = \mathrm{SO}(p)\times\mathrm{SO}(n{-}p)$ where $p = \bigl[\tfrac12 n\bigr]$; when $G=\mathrm{Sp}(n)$ $(n\ge3)$, $K=\mathrm{U}(n)$; when $G=\mathrm{G}_2$, $K=\mathrm{SO}(4)$; when $G=\mathrm{F}_4$, $K=\mathrm{Sp}(3)\mathrm{SU}(2)$; when $G=\mathrm{E}_6$, $K=\mathrm{Sp}(4)$; when $G=\mathrm{E}_7$, $K=\mathrm{SU}(8)$, and when $G=\mathrm{E}_8$, $K=\mathrm{SO}'(16)$.