Ok, I figured it out.  Here is what I did in case it helps anyone else.

On the shopping cart, edit HTML, from the dropdown select Shopping Cart Lines.

<div>
    <t t-call="website_sale.cart_line_product_link">
        <strong t-field="line.product_id.product_tmpl_id"/>
    </t>
</div>

I changed the field in bold so that it now shows [internal reference] Product Name

On the product detail view:

<h1 itemprop="name" t-field="product.name">Product Name</h1>
<span t-field="product.default_code"/>

I added the line in bold so that the internal reference is now displayed underneath the product name.

On the main product page (table view):

<div class="oe_product_cart" t-att-data-publish="product.website_published and 'on' or 'off'">
    <t t-set="product_image_big" t-value="td_product['x']+td_product['y'] &gt; 2"/>
        <t t-call="website_sale.products_item"/>
</div>
<div style="margin-left:10px">
    <t t-if="product.default_code">
        <t t-set="int_ref" t-value="'pn: '+product.default_code" />
            <p t-esc="int_ref" class="text-muted"/>
        </t>
</div>

I added the code in bold. This is the best I could do here.  This will display the Internal Reference at the bottom of each tile.  I would have prefered to show it under the Product Name, but was not able to figure that out.  If anyone has any ideas here that would be great.  I just added the "pn: " to simply prefix the default code with.

Simplest way without coding:

On backend: Website -> Configuration (on top menu) -> Websites -> YourWebsiteName -> Product Page Extra Fields (Tab) ->Add Line (Internal Reference).

Go to product page and there is displayed Internal Reference.

Thats all ;)

Hi Pierre

Were you able to find a solution to your default_code value with variants?