Is “${PS1-}” valid syntax and how does it differ from plain “$PS1”?

The variable expansion ${parameter:-word} will use the value of $parameter if it's set and non-null (not an empty string), otherwise it will use the string word.

Omitting the : will not test if the value is empty, only whether it's unset or not.

This means that ${PS1-} will expand to the value of $PS1 if it's set, but to an empty string if it's empty or unset. In this case, this is exactly the same as ${PS1:-} as the string after - is also empty.

The difference between "${PS1-}" and "$PS1" is subtle, as @Rakesh Sharma notes: both will expand to the value of $PS1, or to an empty string if it is unset. The exception is when set -u is active, in which case expanding unset variables would cause an error. The (empty) default value set by"${PS1-}" circumvents this, expanding an unset PS1 to the empty string without error.

This is standard syntax (originated in the Bourne shell in the late 70s), as are a couple of other, similar expansions.

The syntax:

${parameter:-word}

and a special form:

${parameter-word}

is the valid syntax in POSIX shell, means using the default value word if parameter is unset or null.

Normally, with word is empty, then:

${parameter-}

and:

$parameter

or:

${parameter}

are equivalent.

But under the effect of set -u, all unset variables cause the shell to be terminated. ${parameter-} is used to bypass that strict rule. It works in both case so yes, it's more correct way.