Isn't communicating symbolic thought through drawn symbols that are understood by multiple people the definition of writing?
Proto-writing can be very complex (I remember reading a book where a linguist calls mathematical notation "proto-writing"), but it's not "true writing" until it's capable of communicating essentially any idea you can express in spoken language (it's hard to write "I miss my cat Whiskers" in mathematical symbols) in at least a partly phonetic way (all true writing languages are phonetic to some extent). The earliest examples of that is Sumerian cuneiform and Egyptian hieroglyphs from around 3000 BCE. Whatever this discovery is, it's not true writing.
Early Sumerian symbols depicting kinds of goods (wheat, sheep, beer, etc.), and marks next to these to indicate quantities, are classed as proto-writing.
There's also more general use of symbols to represent ideas or groups, like a cross representing Jesus or Christianity, for example, which aren't classed as writing
I don't think this is the proper definition, since by this standard, Egyptian hieroglyphs, Chinese ideograms, Norse runes and many others would not be considered writing; and any attempt to notate sign languages would not be writing by definition.
Instead, writing is a direct and consistent correspondence between marks and elements of human language (alphabets and abjads represent speech sounds, various ideographic systems represent semantics, you can have hybrids etc). This still makes sure that tallies or just general symbols or icons are not a form of writing, but it doesn't require any phonetic aspect to it either.
There's a spectrum of how phonetic a language is, where Finnish is on one end (sounds very closely align with spelling) and Chinese characters on the other, but all written languages are phonetic to some degree.
The converse is not the case: Chinese is very homophonic so there are a lot of syllable (sounds) that have many different meanings and hence characters associated with them.
It's a common misconception that all Chinese characters are like that, but my understanding is that while there are many, many more ideograms in Chinese than English, something like 80% of the characters do in some way indicate pronunciation (even if it's just something like tone) or use the "rebus principle" or something like that. So again, it's a spectrum, but all writing systems are phonetic to some extent. Human's wouldn't be able to use it to communicate effectively otherwise.
I will say that I'm not a linguist, nor can I read or write a word of Mandarin Chinese, and will happily stand corrected. This is just what I've picked up from reading books about the history and development of writing.
People get confused about them because there's a tie-in with the old Germanic religions where they're used by the gods for divination, and the neopagans have adopted them for that purpose. But they're really just a set of alphabets optimized for carving into wood.
Yes, this is what I had in mind by saying "speech", but you're right, the connection to language is the essential part, and sound just happens to be the paradigmatic medium of human language