I'd like to create a custom Maintenance Page for my needs, but not only when wordpress is updating. I just want to close my site for a period with all the necessarily status code sent. What is the best way to do it?
For such maintenance mode, I found the following plugin a better one (with currently rated a 4.1 on 5):
Easy Maintenance Mode — WordPress Plugins
It's a good one, and the screenshots are there to describe its look and functionality. It'll create a maintenance banner foo all other users except the Admin. It's has a good user connectivity with the social networking links below, so the user who are seeing the Maintenance notice, won't go away - they will be in touch with your social updates.
And there are many facilities described in the plugin page.
BTW, if you find difficulties functioning any plugin like this, try switching to the default theme (Generally TwentyFourteen), and try deactivating all other plugins and activate them one by one to find which one is conflicting. It may help to debug the plugin's conflict.