Cable tear, home button not working. Correct replacement part?

I replaced the battery in my iPhone. After assembling, I got the message the touch ID was not working. Disassembled and found this

To repair this, I need to buy a new display, correct? The displays I’ve looked at here don’t have the home button preinstalled. Am I correct in this assumption? Should I look for a display that has the home button installed or try to remove mine and install in new display?

Edit: After looking at the parts listed on iFixit, I will need to remove my home button and install in new display. Are all displays created equal? There are two on the website for different prices. Is there a better source for displays?