What role has bitcoin-init as a process?
With the order, I will try to response to your question.


  According to this thread I assume that the daemon process bitcoind has been renamed to bitcoin-init but I am not sure if I understood that correctly.


I don't think the bitcoind is rewriting to bitcoin-init, but I don't have a mode for testing bitcoin-core on raspberry, so I ask you if you can post "how you have installed the bitcoin-core?", with command sudo apt install or a zip directory on Github/official site bitcoin?


  My node seems to update the chain, but very slowly - 50GB after several weeks. And it is not reachable if I check my local IP via Coindance.


The answer is "depends", depends on your connection, in any case, is a little slow if you don't have an ADSL 7 mbs


  EDIT: The book (Antonopoulos) sais that bitcoind will give an error unless I set an rpcpassword in the configuration file. I don't have such a file, but bitcoind gives:


Is possible to configure the node bitcoind with the file config, "bitcoin.conf", you have a complete example here and here file must be inside the directory bitcoin, if you don't change it, will be in the folder /home/yourName/.bitcoin/bitcoin.conf or you can set the bitcoind from the command line, an example bitcoind -rpcuser=vincent


  I seem to have two seperate directories for .bitcoin. How can I tell bitcoind to use the other one?


If you have synchronized the bitcoin-core inside another directory (an example because you have a version bitcoin inside the external HDD) your command for bitcoin is similar this, bitcoind  -datadir=/media/vincenzo/Maxtor/BitcoinCore/node if you not call the bitcoind with the daradir, the bitcoind re-synchronized the blockchain of bitcoin inside the /home/yourName/.bitcoin/ directory 


  If I run bitcoin-cli getinfo with my usual Rasperry user pi I get


The problem should be the datadir different from /home/yourName/.bitcoin/