Refreshing only DataTable after button click

I have not used DataTable much but I believe they are used in order to provide various functionalities such as sorting etc easily to your HTML tables. (Reference)

Coming to the problem here, I believe your scope is to refresh the content of a table based upon the AJAX response without having to refresh the whole page.

You can do it using append() and remove() functions of jQuery. A simple algo would be like this:

1. First of all, you can have a row HTML sample somewhere in your page hidden like this:

<div id="definition-row-sample">
    <tr class="definition-row">
        <td class="definition-id"></td>
        <td class="definition-name"></td>
        <td class="definition-description"></td>
        <td class="definition-delete"></td>
    </tr>
</div>

2. Make the .definition-row hidden so that it doesnt show up in your page.
3. Now when you fetch the data via AJAX, first remove all the rows with: $('.definition-row').remove() and then loop through your data and append each row in your table:

    $.each(data, function (i, item) {
        $('#definition-row-sample').find('.definition-id').text(data[i].id);
        $('#definition-row-sample').find('.definition-name').text(data[i].definition_name);
        $('#definition-row-sample').find('.definition-description').text(data[i].description);
        //then simply append it to the table
        let content = $('#definition-row-sample').find('.definition-row').clone();
        $('#projDiscComTable').append(content);
    });

And as for delete, you can define a normal function like:

function definition_delete(id) {
    $.ajax({
        url: 'delete_project_definition',
        type: 'GET',
        data: {
           project_def_id:id
        }
    });   
}

document.on('click', '.definition-delete', function(){
    let id = $(this).parent().find('.definition-id').text();
    definition_delete(id);
});

I have not tested the code but I am sure you get the idea and make it work as I have used similar logic to one of my projects as well.

How about just refreshing the table, after inserting the comment.?

$('#projDiscComTable').DataTable().ajax.reload();

$.ajax({
            url: 'post_project_definition',
            type: 'GET',
            data: {
                discID: $("#disciplinesProjDef").val(),
                discComment: $("#txtDiscipComment").val(),
            },
            success: () => {
                $('#projDiscComTable').DataTable().ajax.reload();
                swal({
                    type: 'success',
                    title: 'Your comment has been added successfully'
                });
            }
        })

Not sure if this fits your requirement but you can still refresh it after inserting the comment.

I figured it out, I redrew the table in the success in the ajax and it seems to work from what I can see. But thank you for all the answers. I will use what you guys have suggested in future.

var ProjDefTable = [];
$('#btnDiscipCommPost').on('click', () => {
    $.ajax({
        url: 'post_project_definition',
        type: 'GET',
        data: {
            discID: $("#disciplinesProjDef").val(),
            discComment: $("#txtDiscipComment").val(),
        },
        success: () => {
            swal({
                type: 'success',
                title: 'Your comment has been added successfully'
            });
            $.getJSON('selectProjectDefinitionDescription', function (data) {
                $.each(data, (i) => {
                    ProjDefTable.push([data[i].id,data[i].definition_name, data[i].description, 
                    "<input type='button' class='btn btn-danger' id='" + data[i].id + "delete' value='-'></input> <span id='" + data[i].id + "deleted' style='display: none; font-weight: 700;'>DELETED</span>"+
                    "<script>"+
                    "   $('#"+data[i].id+"delete').on('click', () => {" +
                    "       $.ajax({"+
                    "           url: 'delete_project_definition',"+
                    "           type: 'GET',"+
                    "           data: { "+
                    "               project_def_id: '"+data[i].id+"' "+
                    "           },"+
                    "           success: () => {"+
                    "               var x = document.getElementById('" + data[i].id + "delete');" +
                    "               x.style.display = 'none';" +
                    "               var y = document.getElementById('" + data[i].id + "deleted');" +
                    "               y.style.display = 'block';" +
                    "           } "+
                    "       });"+
                    "   });" +
                    "</script>"
                    ]);

                    var table = $("#projDiscComTable").DataTable();
                    table.clear().draw();
                    table = $("#projDiscComTable").DataTable({
                        data: ProjDefTable,
                        columns: [
                            {title: "ID"},
                            { title: "Discipline Name" },
                            { title: "Discipline Description" },
                            { title: "Delete" },
                        ],
                        destroy: true
                    });
                })//end else
            });
            var ProjDefTable = [];
        }
    })
});