Several papers have made this distinction, and a few indeed use different terms to distinguish between them. For example, Kazaux et al. (2016) acknowledge that:
These constraints favour the use of a version of the de Bruijn Graph (dBG) dedicated to genome assembly – a version which differs from the combinatorial structure invented by N.G. de Bruijn.
Kingsford et al. (2010) also recognise the distinction:
Note that this definition of a de Bruijn graph differs from the traditional definition described in the mathematical literature in the 1940s that requires the graph to contain all length-k strings that can be formed from an alphabet (rather than just those strings present in the genome).
The oldest reference I found for a specific term to refer to the assembly-related structure is Skiena and Sundaram (1995), where they call it a subgraph of the de Bruijn digraph. Later, in 2002, Błażewicz et al. will refer to it as a de Bruijn induced subgraph. The term de Bruijn subgraph is also formally defined in Quitzau’s thesis (2009). There, and also in the article (Quitzau and Stoye, 2008) the authors describe the sequence graph as a modification of the sparse de Bruijn subgraph (commonly used in assembly problems), where non-branching paths are replaced by a single vertex. The term sparse de Bruijn graph is also used by Chauve et al. (2013).
Another term that I found was word graph, described by both Malde et al. (2005) and by Heath and Pati (2007) as a subgraph or as a generalization of a de Bruijn graph. Rødland (2013) summarises some of the terms used for this data structure:
The data structure is best understood in terms of the de Bruijn subgraph representation of S[k]. (...) Some authors may refer to this as a word graph, or even just a de Bruijn graph.
Although we can recognise that the distinction is not very relevant, the question is asking specifically for the situation where one wants to make such a distinction.
n(mathworld.wolfram.com/deBruijnGraph.html). Q2: yes, implementations don't need all nodes; de Bruijn graph is an abstract entity, a combinatorial structure, like a "complete graph". But if my very important graph miss some edges (b/c useless) I can't call it "complete". It doesn't make it less important BTW! Q3: that's true! Thanks for editing the question. $\endgroup$