I'm now using a workaround by appending a fake line to the output; this way I can rely on the fact that one line will always be present, and this way the empty line is easily distinguishable from the one-line:
r=$(find var/ -name var -printf "%p\n"; echo '==')
# appending additional line to be able to distinguish 0 from 1 found file
case "$(echo "$r" | wc -l)" in
1) # means 0 found files
...
2) # means 1 found file
...
*) # means more found files
...
esac