When I drag e drop the “variable creator” node from the knime paga nodes I have the message
Then I choose Yes to search and install the “knime base nodes” extension and receive the error "knime base nodes"its already installed as you can see beneath
As @HansS pointed out, this very node is not available for KNIME 4.3.4, but was first introduced with KNIME 4.4. Thus, the extension you are installing does not contain this node.
The reason why the KNIME Community Hub is suggesting you to install the extension is because it does only support the latest version of KNIME and has no knowledge about your current KNIME version and with which version of KNIME a node was introduced first.
So in short: In order to reach your goal, you need to update to KNIME 4.4 or find an alternative way to set flow variables (e.g. via a Java Snippet).