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Obviously if the Shel Yad Strap is fully torn then the Tefillin are no longer Kosher, but what if it’s a partial rip? Is there a distinction based on how much of the width is torn? Or is a rip a rip?

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If "tear" still goes over the minimum width (What is the Minimum Width of Tefillin Straps?) it's fine. Otherwise no.

See https://www.thesofercenter.com/case-studies/ripped-retzuah :

"Retzuos start out at 15mm in order that if they begin to tear you will still have some time to get them replaced before they become Possul. The preferable (Kosher Lechatchilah) width is 11mm according to the Chazon Ish and 10mm according to Rav Chaim Naeh. The minimal (Bedieved) width for making a Brachah is 9mm. If a Retzuah started tearing at one spot making the connected Retzuah it less than the minimal width it is Possul and must be replaced."

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    You found a website of someone that sells replacement straps as a source, but it seems to disagree with the Shulchan Arukh, aruch HaShulchan and more in your first source. Probably not the most reliable source one could find... – Double AA Jun 6 at 20:49

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