Which DNA elements belong to the definition of a gene?
Usually a promoter is not considered a part of the gene. Distal regulatory elements qualify even lower for being considered a part of a gene because they can regulate many genes simultaneously.

Basically, a geneic region starts from the transcription start site (TSS) and ends at the terminator (or polyadenylation/cleavage site in eukaryotes). If a gene gives rise to multiple transcripts, those which have similar products (no frameshifts) but different TSS/terminators, we should consider the geneic region as a union of all the primary transcripts produced from that site.

There is a relatively old article on this topic: What is a gene, pose ENCODE? History and updated definition. It summarizes that:


  There are three aspects to the definition that we will list below,
  before providing the succinct definition:  
  
  
  A gene is a genomic sequence (DNA or RNA) directly encoding
  functional product molecules, either RNA or protein.
  In the case that there are several functional products sharing
  overlapping regions, one takes the union of all overlapping genomic
  sequences coding for them.
  This union must be coherent—i.e., done separately for final protein
  and RNA products—but does not require that all products necessarily
  share a common subsequence.
  
  
  This can be concisely summarized as:
  
  The gene is a union of genomic sequences encoding a coherent set of potentially overlapping functional products.


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My summary is that a gene, if it codes for a protein, would be a union of all primary transcripts that produce identical proteins (originating from transcripts with different TSS/terminators) or splice variants.
Ideally, all the different transcripts produced from the locus would have the same promoter.

Overall, it appears that a common promoter is imperative for considering the different products under the same gene because if two products are produced from different promoters then we consider them as products of overlapping genes.

Whether we should consider the promoter as a part of a gene is an open debate.