I'm curious about the general case, but for ease of explaining lets just take the case of a $P^2(\Omega)$ approximation. For simplicity, let's also just consider the reference element $(0,0), (0,1), (1,0)$. To satisfy the global continuity condition, each edge must have three points on it. Thus to have a conforming FE we must use the vertices as three of the six mesh points.
The main question then: Where can the other three mesh points be placed without ruining anything? Clearly, the midpoint of each edge is the standard choice. Similarly, it is clear that each edge must have one point. We can write the last three points as $(0,\alpha_1), (\alpha_2,0),(1-\alpha_3,\alpha_3)$. For what values, if any, of $\alpha_1,\alpha_2,\alpha_3\in(0,1)$ does the finite element posses the following:
- The finite element is conforming (For a second order elliptic BVP).
- The finite element has optimal interpolation properties.
- Nothing else in the FEM I haven't thought of breaks.
Note: I'm already aware that doing as I described makes it impossible to use one reference element for the whole mesh. This does not strike me as a large downside--because for reasonably low order elements it would still be possible to hard code several parent elements with hardly any additional computational cost.
A reference and brief comments would be an ample reply. I've been unable to find any information on this. Every FE text I have encountered simply chooses the standard mesh points and moves along.