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'] > 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.
