Is this a bug in Total?

Question

The output from the last of these lines doesn't make sense to me
Total[{}, {1}]
Total[{}, {1, 2}]
Total[{{1}, {1, 1}}, {2}]
Total[{}, {2}]

0
0
{1, 2}
0

The 3 first correctly sums at the given levels, but the last sums at level 1 unexpectedly. Is this a bug?
My alternative to get the correct output:
total = Fold[Apply[Plus, #, {#2}] &, #, Range @@ MinMax[#2 - 1]] &
total[{}, {1}]
total[{}, {1, 2}]
total[{{1}, {1, 1}}, {2}]
total[{}, {2}]

The thing is {} doesn't contain anything with Depth 2, so there are no entries to sum, so I would expect the structure to be preserved at the levels below level 2. That's what happen at part 1 when running Total[{{}, {{2}}}, {3}] since part 1 is {} which has depth less than 3, there is nothing to sum (not even an empty set) so it's left with its structure in the output {{}, {2}}.
How can you expect 0 from Total[{}, {2}], but not expect {0, {2}} from Total[{{}, {{2}}}, {3}]?

Michael E2 · Answer

FWIW, for the question what to expect, this seems equivalent for the use-case Total[expr, {k}] the OP asks about:
ClearAll[myTot];
myTot[expr_, 0 | {0}] := expr;
myTot[{}, _] := 0; (* the controversial rule *)
myTot[expr_, {k_Integer}] := With[{
    res = Check[
      Apply[Plus, expr, {k - 1}], (* main equivalence *)
      $Failed, {Thread::tdlen}]},
   res /; res =!= $Failed
   ];
myTot[expr_, _Integer | {_Integer, _Integer}] := "Unimplemented";

Examples:
expr = {{1, 4}, {{}, {}}, {10, 20}};
k = 1;
Total[expr, {k}]
myTot[expr, {k}]

(*
  {{}, {}}
  {{}, {}}
*)

expr = {{1}, {10, 20}};
k = 1;
Total[expr, {k}]
myTot[expr, {k}]
k = 2;
Total[expr, {k}]
myTot[expr, {k}]

Total::tllen :  Lists of unequal length in {{1},{10,20}} cannot be added. >>
(*  Total[{{1}, {10, 20}}, {1}]  *)

Thread::tdlen :  Objects of unequal length in {1}+{10,20} cannot be combined. >>
(*  myTot[{{1}, {10, 20}}, {1}]  *)

(*
  {1, 30}
  {1, 30}
*)

expr = {{}};
k = 1;
Total[expr, {k}]
myTot[expr, {k}]
k = 2;
Total[expr, {k}]
myTot[expr, {k}]
k = 5;
Total[expr, {k}]
myTot[expr, {k}]

(*
  {}
  {}

{0}
  {0}

{{}}
  {{}}
*)

Is this a bug in Total?

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