Hi,
If there was no customization to the cart module, you should not have the file templates/YOUR_TEMPLATE/html/com_hikashop/product/cart.php at all on your website.
Renaming that file means that the cart module uses the normal display again. And since you have the "mini cart" option turned off in the cart module options, it's normal that you see the full cart there.
This even indicates that your cart module view had been heavily customized as it is not taking into account that setting of the module.
So turn on the "mini cart" option of the cart module and it will look a bit better and the link to the checkout will work.
Now the display will probably be different from what you had since by renaming that file, you basically remove all the customization done to the cart module.
I suppose that you're not the one who did that customization (or you would have been able to fix that link yourself) and thus the best would be to contact the person who did that (if you didn't hire someone to do that for you, then it means that this customization was part of your Joomla template and so it's the support of your Joomla template that you want to contact), so that he can fix the link in his customization of the cart module.