Cannot convert '1/13/2013' to date value

You can use the optional culture argument with Table.TransformColumnTypes(). Using "en-US" will interpret the text as "mm/dd/yyyy" whereas other cultures like "en-IN" will interpret the text as "dd/mm/yyyy".

For example, this should shouldn't throw an error:

let
    Source = #table({"Date"}, {{"1/13/2013"}}),
    #"Changed Type" = Table.TransformColumnTypes(Source, {{"Date", type date}}, "en-US")
in
    #"Changed Type"

 It would if you used "en-IN" instead.