Left, Right, DoubleClick functions

I've been trying for a while to separate different types of click onto a mouse. Originally I used to look at which mouse button is clicked, and then call a different function for each one, and then check if it was double click or not, but its a bit of a mess. What Id like to do is just add a function via a bind, as in:-

$("#divNY").on("click", function() {
  $("#divNY").html("Single Click");
});
$("#divNY").on("rightclick", function() {
  $("#divNY").html("Right Click");
});
$("#divNY").on("dblclick", function() {
  $("#divNY").html("Double Click");
});

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If there was a div called 'divNY', like:-

<div id="divNY">nothing yet</div>

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Reading up on things, I know I need to add a rightclick function in like to use contextmenu as well:-

(function($) {
  $.fn.rightclick = function(method) {
    $(this).bind("contextmenu rightclick", function(e) {
      e.preventDefault();
      method();
      return false;
    })
  };
})(jQuery);

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Ive tried to just use the below, but it doesnt work:-

$("#divNY").on("contextmenu", function() {
  $("#divNY").html("Right Click");
});

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Anyone know how I can use my code like:-

$("#divNY").on("click", function() {
  $("#divNY").html("Single Click");
});
$("#divNY").on("rightclick", function() {
  $("#divNY").html("Right Click");
});
$("#divNY").on("dblclick", function() {
  $("#divNY").html("Double Click");
});

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Ive put my work soo far on a jsfiddle at - https://jsfiddle.net/m3yfrt7q/

Huge thanks in advance :-)