Looking at https://jmeter-plugins.org/wiki/GraphsGeneratorListener/ plugin it mentions server "hits". Is that when JMeter wrote HTTP request to the Java socket, or does that include the server's reply? What about network errors, like the request has been sent, but no reply received within the sampler's request timeout, or there was an established connection error, or connection failed to establish - is that still a hit?
From source code, it looks like Server Hits per Second checks sample start time:
private void addHits(SampleResult res) {
// ...
addHit("Server Hits per Second", normalizeTime(res.getStartTime()), 1);
Unless I missed something, it's seems unaware of the protocols at all: it will build the graph based on start time of all samplers/sub-samplers included in Graphs Generator configuration, regardless of their type. That includes filter on their status (failed or success).
So answers to your questions depend on Graphs Generator configuration: you can include and exclude failed results, which will define whether timed out or connection error samplers will be included or excluded. To achieve "JMeter wrote HTTP request to the Java socket" you need to make sure only HTTP requests are included. To only include requests which received some response, you need to exclude failed requests.