There is a big difference between using a single pipe (pipe output from one command to be used as input for the next command) and a process control OR (double pipe).
cat /etc/issue | less
This runs the cat command on the /etc/issue file, and instead of immediately sending the output to stdout it is piped to be the input for the less command. Yes, this isn't a great example, since you could instead simply do less /etc/issue - but at least you can see how it works
touch /etc/testing || echo Did not work
For this one, the touch command is run, or attempted to run. If it has a non-zero exit status, then the double pipe OR kicks in, and tries to execute the echo command. If the touch command worked, then whatever the other choice is (our echo command in this case) is never attempted...