Deleting duplicates only if they are a certain kind of duplicate

list = {{1, a, b}, {1, 1, b}, {1, a, a}, {b, a, 1}, {b, a, b}};
l1 = Select[list,DuplicateFreeQ[DeleteCases[#,1]]&]

{{1, a, b}, {1, 1, b}, {b, a, 1}}

DeleteDuplicatesBy[Sort]@l1

{{1, a, b}, {1, 1, b}}

list2 = {{1, 1, 1, 1}, {1, a, a, b}, {a, b, b, c}, {1, a, b, c}, {1,  1, a, b}};
l2 = Select[list2, DuplicateFreeQ[DeleteCases[#, 1]] &] 

{{1, 1, 1, 1}, {1, a, b, c}, {1, 1, a, b}}

DeleteDuplicatesBy[Sort]@l2

{{1, 1, 1, 1}, {1, a, b, c}, {1, 1, a, b}}