To answer your information from the comments (which actually should be edits to the question):
So far, it has been very easy to use WP to replace my self-made code used to manage users’ login, write and output posts and static pages in 3 of my existing websites, without altering anything else the websites.
You wouldn’t believe how easy it is to put some random code in WordPress context: Use a plugin, a theme or a mu-plugin (or if appropriate a Drop In). And you wouldn’t believe how less work it is:
<?php /* Plugin Name: I AM DROPPING MY CODE HERE */
// Load your "out of WP context"-code here
is all you need. Really!
Just stuff your code in a folder, put it in the ~/wp-content/plugins/ folder in a subfolder named of your choice and put above code in the file that loads your routines. That’s it.
I was hoping to leverage WP a bit more by customizing its admin-bar and replace the one I currently use.
Then use a (mu-)plugin. If you are now trying to build a DB connection and other stuff, then you will anyway simply load WordPress like WordPress loads itself. The thing you will end up is a not maintainable wrapper for WordPress, while WP delivers convenient wrappers already by itself in the form of (MU-)Plugins & Themes.
Hint: A theme isn’t really much more effort than a plugin. Just add a folder to your themes folder (name of your choice) and two files: index.php and style.css. As soon as you add a Comment on top of your style.css file you have a theme that you can activate in the back end. The index.php will hold your code and be loaded on every request.
/* Theme Name: Twenty Thirteen */
I will look into plugins and how that could let me use my code in a standard WP installation. But as always I have little time, learning too much about WP is not the short term solution I am looking for.
You already invested quite some time to ask this question and try to work around WP. All the resources you will need to proceed further will be useless for you as long as you are out of context of WP. This is a simple calculation: 10 minutes to bring your code in context vs. whatever time you needed to ask this question, comment, etc. and all the pain you will suffer from any next task.