Please Learn from this input.
Yes, you are in fact doing what you state you don't do: "Our company does not print packing slips from invoice"
This step is Preparation of the invoice: "rather we use the Order Fulfillment feature."
It's simply a different perspective of the same concept, which allows you to see a cross-reference for all orders, all customers, that same item, its fulfillment potential, etc.
"When printing the Packing Slip, we want only the items that are in the package to print. Right now it is printing the entire order and, if applicable, the tax code line if the customer is taxable."
What you need to do is Open the Lists menu > Templates list. Here you see two pieces of info: Template Name and Template Type. You need to understand if your Packing slip is a Version of the Invoice Template. The same is true for Pick List. You see that the Name of the template relates to the transaction type that you are creating. An Invoice is On Screen, as create Invoice; you see in the Print options, to print it on a Design or Template that you linked to the function for Packing Slip or Pick List. That Default Function is controlled in your Edit menu > Preferences, Sales and Customers, Company Preferences. Here is where you control the Print Default that is the bottom left of the Fulfillment screen, and Top, the Print icon for "print packing slip" or Print invoice.
Notice in my attachment that there are Two Packing Slip options: from Sales Order or from Invoice as Design options.
Notice also on the Far Right, the option to "Don't print items with Zero Amounts." You see them on the screen, but that doesn't print for the customer output view. You need to set Preferences to meet your specific needs.
For purposes of showing Sales taxes or any other columns, that is your choice of various Templates for that one Transaction Type. I can make an invoice, then choose the Template from the Template Dropdown, and not using the Print Menu, to make an invoice that doesn't show specific things per each of the various Templates I might want to use. That means Changing the Template while the transaction is open on my screen, then Printing with the Print icon, not selecting from the Print icon's own menu.
You use the Fulfillment Worksheet to Fulfill. You can control everything else from the Settings and the Invoice once you open it on the screen, if you need more control than from the Preferences that are used by the fulfillment worksheet, which you are supposed to set up Preferences.
There are over 400 settings in Preferences, by the way, that you should control.
Please see my attachment.