To answer your question it will help with a little network background.
By and large, the internet is built as a series of clients (e.g. your MacBook Pro), servers (e.g. Netflix, Google, etc.) and network providers (e.g. your ISP) who link them all together.
Whenever you do anything on the internet, your client (MacBook Pro) sends a request to the server, which then sends back a reply. There may be many network providers in the middle of that connection that handle the routing of that request.
What network Diagnostics does is look at all the steps to try and determine where the failure is.
In Network Diagnostic's terms:
Network Settings - this is your client's configuration - is it setup correctly to talk to the internet
ISP - this is the network provider that you connect to (and pay)
Internet - this is the intermediate network providers between your ISP and...
Server - the actual server/company you're trying to interact with.
Now, from what you're saying, your ISP connection stays green and the problems tend to be 'upstream' from there... i.e. in the Internet and Server levels. Unfortunately these are almost completely outside of your control.
The best solution I can offer is to contact your ISP's support department and report your experience. It is *possible* that the problem is inside their network, where they connect to other ISPs (e.g. it could be their links are saturated and causing the drops), and if that's the case they should be able to do something about it. If the problem is upstream (in some other network) then the ISP may be able to work out a solution
If the problem is at the server itself then contacting the site operator might be in order, but I'm guessing this isn't just one site that's affected, which tends to point away from it being the server-side that's at fault.