First, you don't need multiple document.ready functions. Everything you want to do in response to the document ready event should usually be within a single function.

Second, even if you have multiple signature fields, there is still only one modal used to collect a signature so both of the events you have are targeting the same element. Basically, even if they're working it would change twice on document ready and the last one to fire would stick.

As a result, what you need to do is change it dynamically based on which button is clicked.

A few other notes:

• You could hide the entire signature nav since you don't need them to make a choice
• You can hide the Type/Draw options with CSS instead of JavaScript; both are fine, but I usually prefer CSS for that kind of thing
• The .find() method isn't needed because you can configure the CSS selectors to target those buttons directly without using find

CSS options for hiding the modal tabs/nav (only need one)

/* hide entire signature modal nav */
#sigNav {
  display: none !important;
}

/* hide draw nav link only (using attribute selector) */
#sigNav a[href="#drawTab"] {
  display: none !important;
}

/* hide draw nav link only (using index selector) */
#sigNav li:nth-of-type(2)  {
  display: none !important;
}

JavaScript to change the values based on which button is clicked

$(document).ready(function(){
  // neither of these are needed if you use the CSS option
  //$('#sigNav').hide(); // hide entire menu
  //$('#sigNav a[href="#drawTab"]').hide(); // hide just the draw option
  
  $('#form-signature-dlg').on('shown.bs.modal', function(){
    $('#sigNav a:eq(0)').click();
  });
  
  /*Initial button label*/
  $('.myInit .Sign_Sig').val('Initials'); // .find() is not needed
  
  /*Update modal on click event for initials*/
  $('.myInit .Sign_Sig').on('click',function(){
    updateSignatureModal('Initial','initials');
  });
  
  /*Signature button label*/
  $('.mySign .Sign_Sig').val('Signature'); // .find() is not needed

  /*Update modal on click event for signature*/
  $('.mySign .Sign_Sig').on('click',function(){
    updateSignatureModal('Sign','name');
  });
});

// reusable function for updating the modal
function updateSignatureModal(type, input){
  var title = type + ' Document';
  var placeholder = 'Type your ' + input + ' here';
  
  $('#sig-dlg-title').text(title); // header
  $('#form-signature-dlg .signature-btn').text(type); // button
  $('#typeSignature').attr('placeholder',placeholder); // placeholder
}

Technically, you can update the button value and attach the event handler together by stringing the methods together. I kept them separate for clarity, but you could do the following as well.

$(document).ready(function(){
  $('#form-signature-dlg').on('shown.bs.modal', function(){
    $('#sigNav a:eq(0)').click();
  });
  
  /*Initials button label and set click event to update modal*/
  $('.myInit .Sign_Sig').val('Initials').on('click',function(){
    updateSignatureModal('Initial','initials');
  });

  /*Signature button label and set click event to update modal*/
  $('.mySign .Sign_Sig').val('Signature').on('click',function(){
    updateSignatureModal('Sign','name');
  });
});

// reusable function for updating the modal
function updateSignatureModal(type, input){
  var title = type + ' Document';
  var placeholder = 'Type your ' + input + ' here';
  
  $('#sig-dlg-title').text(title); // header
  $('#form-signature-dlg .signature-btn').text(type); // button
  $('#typeSignature').attr('placeholder',placeholder); // placeholder
}