By calling click() method you are programmatically triggering the click event. Need to change your code to delegate click to call your code.
Example:
$("#UiLogo").click(function() {
alert("1 - Called");
$("#UiLogo").val(data.result.Value);
$("#ImgUiLogo").css('display', 'block');
$("#ImgUiLogo").attr("src", JsLogoPath + data.result.Value);
});
$("#AdminLogo").click(function() {
alert("2 - Called");
$("#AdminLogo").val(data.result.Value);
$("#ImgAdminLogo").css('display', 'block');
$("#ImgAdminLogo").attr("src", JsLogoPath + data.result.Value);
});