After some research on the net I found serial examples for serial communication with the STM board here and here. To bring an important information in advance: the second exmample uses the Open Source library libopenmc3. I experimented with their given USART example a lot and sending data was successfull. Also receiving data worked fine. But after enabling the TX_RX mode, I was not able to send data anymore. Perhaps someone has a solution - I do not have onee so far. That is why I than chose the default library from STM and was had success with the first mentioned from the web - with sending and receiving data!
The following images show only "sending" data from the STM board to the Mac / PC [Update: Receiving data is solved too now (without libopenmc). But later more to this].
To check if this works I finally could use my Logik Analyser. Fro all who are interested in this gadget, it is available from Saleae and has nice analysis features. And the best: The software for this nice tool runs on Linux, Windows and Mac OS X!
The serial example now sends a character from the STM board. These serials signals can be measured on the corresponding pins on the board. Please remember that your must not connect these pins directly to a PC or Mac: The STM-Board works with a 3.3 Volt level for "logical HIGH"! A PC or Mac works instead of this with 5 or even 15 Volts!
And this is how the test set-up looks like:
This is the relevant code snippet from the source code:
This is a screenshot from the software of the Logic Analyser which recognises the serial protocol as well (if selected in the software). You can even see the sent characters: