You are not required to use the Partial.ly checkout button on your website. Adding the basic code snippet from the integration tool will automatically add the Partial.ly checkout button to your website.
If you'd like to use your own image, just select the 'Use a custom image for the checkout button' checkbox on the integration tool and add the image url for the image file that you'd like to add.
It is recommended to use the Shopify files section of your Shopify admin to add the image. Upload the file and then copy and paste the resulting code in the image URL field on the integration tool for the offer you want to add to your site.