Definition(s)/characterization(s) of intractability
What is intractable or not inherently depends on what the underlying computational model is as well as what your conventions are. By far, the most widely used notion is based on the Cobham-Edmonds thesis, which states that, for any given machine model for which time (or any equivalent thereof) is a valid resource, the tractable problems are those that can be solved in polynomial time. In this sense, tractability quite certainly has a formal definition.

This means that, if you are using run-of-the-mill TMs, then the class of tractable problems is $\mathbf{P}$. If you use probabilistic TMs (with two-sided error), it would then be $\mathbf{BPP}$. If you use quantum circuits, $\mathbf{QBP}$. For ATMs, $\mathbf{PSPACE}$—and so on.


  Are there problems/contexts which could be described as being on the border, meaning that they're really not easily fitting in any of the 2 categories (tractable/intractable) ?


The way tractability is usually understood, this is impossible. There is a well-defined set of tractable problems. Using standard logic, a problem is either in that set or it isn't. You could identify some problems as more difficult than others, but that is a different concept altogether since it would produce a whole spectrum of hardness (and, hence, is not quite captured by the aforementioned notion of tractability). What would rather fit that notion is time complexity itself; after all, it allows you to express a problem's hardness based on the time resources (as a function on $\mathbb{R}_0$) needed to solve it. (And, as we know, most such functions can be compared using, for instance, asymptotic analysis.)

TL;DR: Intractability is a discrete concept; time complexity is a continuous one (or as continuous as it can be in the set of (computable) $\mathbb{N} \to \mathbb{R}_0$ functions).