Number of distinct configurations of equiangular lines passing through origin

Question

We have a $d$ dimensional space and want to place $n$ lines passing through the origin in such a way that the angles between any two pairs of lines is the same. Such a placement results in a configuration. Two configurations are considered identical if:

1. We can rotate one and get another.
2. We can start with one configuration, move the origin and get another.
3. Start with one and a combination of the above two leads to the other.

The first one is simple enough. For an example of the second one, consider the case of $d=3$. Now, consider three lines on the $x$-$y$ plane at $120$ degree angles. If we move the origin along the z-axis, we get a pyramid and the equiangular property is maintained. However, all these configurations are considered equivalent to the planar configuration (note the cartesian coordinate system is one of these pyramids). Another way to think of this is a tri-pod stand with origin at the head. You move the head up or down by gradually increasing or decreasing the angle between the legs of the tripod.

As an example of distinct configurations, consider again $d=3$ and $n=4$. We can either take the center of mass of a Tetrahedron (which is at the origin) and draw lines to each of the vertices. The angles between any two of these lines will be $60$ degrees. Or, we could take an Icosahedron and draw any $4$ of the main diagonals. The angle between any two of these is about $97$ degrees. Also, we can't take the first configuration and move the origin to get the second one. Hence, these are at-least two distinct configurations for the case of $d=3$ and $n=4$.

Let's call the number of such configurations $\alpha(d,n)$. We therefore have: $\alpha(3,4)\geq 2$.

Is there a closed form or anything else we can say about $\alpha(d,n)$ in general?

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NOTE:

I asked a very similar question earlier today: Number of configurations of lines in $d$ dimensional space where any pair of them have the same angle.. On asking, I realized via comments an added condition that would make it more interesting (condition #2 above). However, since the original question is still technically valid, I decided to leave it as-is.