What that notice is about will not affect you as you're using Payments Standard - this product uses basic button code which is generated via the online button creator. Bottom line, you're OK.
If you were a developer using Express Checkout, Website Payments Pro, MassPay, or Button Manager and working with APIs that are associated with these products then, yes you would have to make some changes for your code to work properly.
As far as changing your site from http:// (non-secure) to https:// (secure) - that is something to consider. Starting in July with the next big update to Google Chrome, Google Chrome will mark all HTTP sites as ‘not secure’ and prominently highlight this in its URL bar. This may or may not spook potential customers - you know that PayPal takes care of the purchasing security for you and your site is safe however, the customer may not understand that and seeing that your site is not secure may raise some questions. There's a brief explanation here.
