How to reach websites that have failed while offline? When I try to open a website when my connection is down or while my modem is starting up, my Mac flags this url as unreachable. 

When the connection is finally open and internet available, I still can not reach this web site as if the Mac doesn't try to reach it again and still displays me the error page. Other web sites (that I didn't try to reach while offline) are reachable...

Is there a way to reset this information so that the Mac really tries to reach url ?
I don't think you are seeing anything out of the ordinary here. My connection goes down all the time, and it's never been immediate for me, but refreshing does bring the site up faster. 

My theory is that it just takes another amount of time for the browser to sort out that the connection is available again; the browser periodically pings the system to see if there is a connection available, and the lag you're seeing is waiting for the process to run its course again. Perhaps it pings once a minute as opposed to once every few seconds for fear of bogging down the processor.

Computers, as a rule, don't like it when things unexpectedly fail and thus take longer to get back up and running as opposed to when everything goes right.