If this is a persistent HTTP connection, I think the reason for this behavior is that when you refresh your browser, you are just sending a new request within the already existing connection. When that connection was created the CSS assigned it to one of the servers and therefore every request within that connection will go to the same server.
When you close your browser, probably the connection is being RST, and when you open it again a new connection is being established and that is why you see it going to the other server.
This is the way HTTP works and I do not think that you can change this behavior on the CSS.
When you configure no persistence on the content rule, you are instructing the CSS to initiate a non-persistence connection in the backend (to the server).
You can configure the global command "persistence reset remap", so the CSS will RST and remap each request, however this configuration is used when you have many content rules with same VIP and different layer 5 configuration, so you could have different server farms for different URLs, and then you would like to have the CSS checking at every request and remap if a more specific content rule is matched.
I do not think that this configuration would change your behavior with a single persistent connection from the browser to the same rule, but you might want to give it a try.
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