It is pretty clear that with shell scripting this sort of thing can be accomplished in a huge number of ways (more than most programming languages) because of all the different
I see several questions here.
“Can I write something that actually reflects this logic”
Yes. There are a few ways you can do it. Here's one:
if [[ "$1" != "" ]]; then
DIR="$1"
else
DIR=.
fi
“What is the difference between this and DIR=${1-.}?”
The syntax ${1-.} expands to . if $1 is unset, but expands like $1 if $1 is set—even if $1 is set to the empty string.
The syntax ${1:-.} expands to . if $1 is unset or is set to the empty string. It expands like $1 only if $1 is set to something other than the empty string.
“Why can't I do this? DIR="$1" || '.'”
Because this is bash, not perl or ruby or some other language. (Pardon my snideness.)
In bash, || separates entire commands (technically it separates pipelines). It doesn't separate expressions.
So DIR="$1" || '.' means “execute DIR="$1", and if that exits with a non-zero exit code, execute '.'”.