I am not aware of any such notation (and in the business of choice functions, one runs a lot into $\mathcal P(S)\setminus\{\varnothing\}$).
It is fine to make your own, but be sure to be consistent about it, and to define it at the beginning of your work.
There is a risk of having too many notations, it may burden the reader. Sometimes just writing it explicitly works just as well. If you're tired of doing that, write a LaTeX macro.
The empty set is indeed a set (the set of no elements) and it is a subset of every set, including itself. $$\forall A: \emptyset \subseteq A,\;\text{ including if}\;\; A =\emptyset: \;\emptyset \subseteq \emptyset$$
$$\text{BUT:}\quad\emptyset \notin \emptyset \;\text{ (since the empty set, by definition, has no elements!)}$$
That is, being a subset of a set is NOT the same as being an element of a set: $$\quad\subseteq\;\, \neq \;\,\in: \;\; (\emptyset \subseteq \emptyset), \;\;(\emptyset \notin \emptyset).$$
$\emptyset \;\subseteq \;\{1, 2, 3, 4, 5\},\quad$ whereas $\;\;\emptyset \;\notin \;\{1, 2, 3, 4, 5\},\;$.
$\{3\} \subseteq \{1, 2, 3, 4, 5\},\quad$ whereas $\;\;3 \nsubseteq \{1, 2, 3, 4, 5\}, \text{... but}\; 3 \in \{1, 2, 3, 4, 5\}$.
Best Answer
I would say the following should be not too controversial:
$\emptyset$ and $\varnothing$ are typographical variants of the same mathematical symbol designating the empty set
The symbol was introduced by Bourbaki, was inspired by the Norwegian character Ø, but is a distinct character from it
The intention was most probably to create a symbol related to $0$ (zero), not to O (Oh), distinguished from it by striking it through. After all the empty set has all kinds of relations with the number $0$, but none with the letter O. (By contrast big-Oh and little-o symbols derive from the word "order".)
The symbol has absolutely no relation (apart from appearance) with the lower-case Greek letter phi, with typographical variants $\phi$ and $\varphi$.