Definitions of pyramid (eg, tetrahedron as "$1$-fold $3$-pyramid" vs "$3$-fold pyramid") My lecture note and my textbook offer slightly different definitions of pyramid.
Here's the one from the lecture:

And here's the one from the textbook:

I just want to make sure I interpret the two statements correctly. Am I right to assume that, according to the first definition, a tetrahedron would be called a 1-fold 3-pyramid, while according to the second definition, it is a 3-fold pyramid? 
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Edit 1: the reason for the above question is because in our class we also have a lemma. 

Lemma: Let P be an r-fold d-pyramid over Q, then the number of k-face of P is $f_k(P) = \sum_{i=0}^r \binom{r}{i}f_{k-i}(Q).$

Taking the tetrahedron as an example, this formula only works if we see it as a 1-fold 3-pyramid, but not a 3-fold pyramid.
Edit 2: so far I haven't been able to find a lot of examples of pyramids according to the first definition. If possible please show me a few examples with different r's. 
 A: The wording $d$-pyramid just represents a shorthand for $d$-dimensional pyramidal polytope, that is a tip atop a $d-1$-dimensional base polytope.
Whenever the base itself is a pyramid in turn, then both your definitions speak of a 2-fold pyramid. (Jonathan Bowers, aka PolyhedronDude, then speaks of a scalene.) If the base of that subdimensional pyramid again is a pyramid, then you have a 3-fold pyramid or tetene. Etc.
As an example take the general cross-polytope. The 3-cross-polytope is the octahedron. Chopping off one vertex you get the square pyramid. I.e. a single tip atop a square base. - In terms of your definitions that on would be a 4-gonal 1-fold 3-pyramid.
Within 4-space the cross-polytope is the 16-cell with 8 vertices. If you omit one of its vertices you get the pyramid on top of an octahdral base. If you further omit a second vertex, which is taken from the octahedral base, then you still have a pyramid, but its base deforms into a square pyramid itself. Thus you have a 2-fold pyramid above a square (base-)base. - In terms of your definitions that last thingy with 6 vertices would be a 4-gonal 2-fold 4-pyramid.
In the same way you could derive from the 5-dimensional cross-polytope a 4-gonal 3-fold 5-pyramid with a base square (4-gonal) and 3 mutually orthogonal tip extensions (3-fold), living wthin 5D (5-pyramid).
--- rk
