I suspect your confusion arises in part because homotopies of paths are continuous maps $I^2 \to X$, while 2-morphisms in $\pi_{\lt \infty} X$ are continuous maps $\Delta^2 \to X$. That is, 2-morphisms are not strictly the same as homotopies of paths.
The dictionary between the two structures is not too bad:
A 2-morphism in $\pi_{\lt \infty} X$ is a homotopy of paths, where either the beginning or the end is fixed (or perhaps it is a homotopy to or from a constant path).
A continuous map $I^2 \to X$ can be viewed as a composite of two 2-morphisms in $\pi_{\lt \infty} X$. You end up using the diagonal $(0,0) \to (1,1)$ in the square to separate the two 2-morphisms, because the simplicial structure of $\Delta^1 \times \Delta^1$ has a diagonal 1-simplex.
I personally find it rather magical that among the sixteen 2-simplices in $\Delta^1 \times \Delta^1$, a pair of them pops out as non-degenerate - it is a rewarding computation.
More generally, higher dimensional cubes, viewed as products of the simplicial set $\Delta^1$, have canonical decompositions into nondegenerate simplices. The corresponding higher homotopies are composites of homotopies with various pieces held constant.
I think this should account for the discrepancy you see in the number of face maps.
One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories:
arXiv:1112.0040
(i.e. $(\infty,n)$-categories).
This axiomatization includes several variants of $(\infty,n)$-category for finite n, such as what you mention in your question. In particular, when also taken in context with the comparison results of Bergner-Rezk (which you mentioned) and Lurie arXiv:0905.0462, I would say that there is a clear consensus for what the homotopy theory of $(\infty,n)$-categories should be, and that it is fairly rigid (few automorphisms). It is at least fair to say that such a consensus is forming.
Once you pin down the theories of $(\infty,n)$-categories, I would say that there are two distinct reasonable choices for what the (homotopy) theory of $(\infty,\infty)$-categories should be. So in that sense the answer to your question is no, there is not a single theory of $(\infty,\infty)$-categories; there are exactly two. But this "no" is a far cry from saying that there is a vast uncharted landscape of possibilities.
What are these theories? As Charles Rezk mentions, the inclusion of $(\infty,n)$-categories into $(\infty, n+1)$-categories has both a left and right adjoint and this gives rise to two towers of homotopy theories of higher categories. The limits of these towers give the two potential models of $(\infty,\infty)$-categories. They are not equivalent. Let's call the limit using right adjoints $Cat_{(\infty,\infty)}$ and the limit using left adjoints $LCat_{(\infty,\infty)} $.
One consequence of the unicity result is that these towers are essentially uniquely defined and are essentially model independent (provided the models satisfy our axioms). So in that sense there are these two canonical (established?) choices for the homotopy theory of $(\infty,\infty)$-categories. I know Clark and I have discussed this idea with many people, but the idea is certainly not new.
I think one of the important parts of this story is Eugenia Cheng's theorem from her paper "An omega-category with all duals is an omega groupoid". Her result applies in the tower using the left adjoints, $LCat_{(\infty,\infty)} $, the "coinductive" version. There an $(\infty, \infty)$-category can be tought of as a sequences of $(\infty, n)$-categories, where each truncates to the previous theory. Cheng's result implies that in such a higher category if you have all duals you are an $\infty$-groupoid.
I prefer the other limit, the limit of right adjoints. There an $(\infty,\infty)$-category is a sequence of $(\infty,n)$-categories where the previous is the maximal $(\infty,n-1)$-category. For some reason this seems more natural to me, though I know of others who disagree. This version includes one of my favorite examples: the infinite cobordism category. There are cobordisms and cobordisms between cobordisms and cobordisms between these and so on forever. This is an object in the tower of right adjoints which has duals for all objects, but which is not an $\infty$-groupoid. The limit of right adjoints fails Cheng's theorem. It also has a sort of inductive notion of equivalence, rather than coinductive.
Your idea of taking the union of the theories of $(\infty,n)$-categories is actually not as far off as you might think. One problem with the naive union is that it is not cocomplete. There are diagrams which increase category number and have no colimits, or at least not the colimit which should exist. One way out of this is to note that you probably want the homotopy theory of $(\infty, \infty)$-categories to be presentable. So instead of taking the colimit naively you should take the colimit in presentable $(\infty,1)$-categories. Now since the inclusion functors preserve both limits and colimits, you can take this colimit in either $Pr^L$ or $Pr^R$. In either case you can use the equivalence $Pr^R \simeq (Pr^L)^{op}$ and Higher Topos Theory 5.5.7.6 or 5.5.3.13 to compute these colimits. In short you compute the colimit by taking adjoints and computing the limit naively. Thus you find:
Taken in $Pr^L$:
$$ \mathrm{colim} \; Cat_{(\infty,n)} = Cat_{(\infty,\infty)} $$
taken in $Pr^R$:
$$ \mathrm{colim} \; Cat_{(\infty,n)} = LCat_{(\infty,\infty)} $$
Which is the "right" notion of $(\infty,\infty)$-category probably depends on taste and what you want to do with the notion. Both are useful. The above descriptions gives you a variety of universal properties for these two theories. I would also love to know if there are any theories in between these two. I also strongly believe that $LCat_{(\infty,\infty)} $ is a localization of $Cat_{(\infty,\infty)} $, but I haven't written down a proof. Perhaps it is easy to see this from the above description?
You can be more explicit about these models too. Dominic Verity has a model of higher categories based on "weak complicial sets". There are $(\infty,n)$-versions of this and also $(\infty, \infty)$-versions.
One of the conjectures that Clark and I made at the end our paper is that some variant of Dominic's $(\infty,n)$-theory satisfies our axioms (Dominic, Emily Riehl, and I have a partial sketch of this, so hopefully the truth of this conjecture will be known... soon?). If that is true, then it is straighforward to show that the Dominic's $(\infty,\infty)$-version of weak complicial sets is a model of the tower of right adjoints, the one which includes the infinite bordism category. So there are also concrete models of these theories.
There are also others whose work will yield explicit models of this. Rune Haugseng's work was already mentioned. Jeremy Hahn's (upcoming?) work will provide nice model of both limits. I am sure there are many ways to model these two theories.
So to summarize:
- There are now a variety of uniqueness and comparison results which pin down the theory of $(\infty,n)$-categories as well as it is pinned down for $(\infty,1)$-categories.
- It actually is clear how to send $n \to \infty$; there are two ways to do it, giving two such infinite theories.
- We can describe and study these theories explicitly and they have interesting properties and possess interesting examples.
- Does this constitute a "consensus"? Of course not, but I hope it goes a little towards answering the real question.
Best Answer
Here's an answer for question 1. (This bothered me for a long time too, I also could never find a formal definition in the literature!)
If one uses 'nice' topological spaces (so that Top has an internal hom), then one can define a homotopy from a map $f: X\to Y$ to another map $g:X\to Y$ to be a continuous map $$H:[0,1]\to\mathbf{Map}(X,Y)$$ with $H(0)=f$ and $H(1)=g$.
Then, given two such homotopies $H_1,H_2:[0,1]\to\mathbf{Map}(X,Y)$, we can define a homotopy of homotopies or a $2$-homotopy $\eta: H_1\to H_2$ to be a continuous map $$\eta:[0,1]\to\mathbf{Map}([0,1],\mathbf{Map}(X,Y))$$ with $\eta(0)=H_1$ and $\eta(1)=H_2$.
One can keep doing this, defining a notion of $n$-homotopy using that of an $(n-1)$-homotopy.
To get the usual definition (which works for arbitrary topological spaces!), one uses that the functors $(-)\times X$ and $\mathbf{Map}(X,-)$ are adjoint.
So a homotopy $$H:[0,1]\to\mathbf{Map}(X,Y)$$ is the same as a continuous map $$H^\dagger:X\times[0,1]\to Y,$$ where the endpoint conditions are now given by \begin{align*} H^\dagger(x,0) &= f(x),\\ H^\dagger(x,1) &= g(x) \end{align*} for all $x\in X$.
Similarly, a homotopy between homotopies $$\eta:[0,1]\to\mathbf{Map}([0,1],\mathbf{Map}(X,Y))$$ is the same as a map $$\eta^\dagger:[0,1]\times[0,1]\to\mathbf{Map}(X,Y)$$ such that \begin{align*} \eta^\dagger(0,t) &= H_1(t),\\ \eta^\dagger(1,t) &= H_2(t) \end{align*} for all $t\in[0,1]$, which is the same as a map $$\eta^\ddagger:[0,1]\times[0,1]\times X\to Y$$ such that \begin{align*} \eta^\ddagger(0,t,x) &= H^\dagger_1(t,x),\\ \eta^\ddagger(1,t,x) &= H^\dagger_2(t,x) \end{align*} for all $t\in[0,1]$ and all $x\in X$.
One can keep going like this for $n$-homotopies, applying adjointness $0$ to $n-1$ times, leading to $n$ equivalent definitions of an $n$-homotopy!