First, a conversation is not a legal process - so it's unclear what you mean by "legally binding".
Now, there could be reasons why what was said in a conversation is legally important and might be something that someone would want to introduce into evidence in a legal proceeding.
However, on their own, personal notes are hearsay - out-of-court statements that were not made under oath - and are prima facie inadmissible. The testimony of what was said at the meeting by one or more of the participants who are available for cross-examination is admissible evidence if relevant. That testimony is the primary evidence of what was discussed.
Now, if there is a dispute over what was discussed, contemporaneous notes may help the trier of fact to decide which version they believe, if deemed admissible, but they are only one of many factors considered. However, because they only have one author, while they might be a full and accurate record, they could also be a complete work of fiction. They could also be post-hoc forgeries - i.e. not actually created at the same time - the onus is on the person seeking to introduce them to prove they are what they purport to be.
Formal minutes of the meeting that are distributed and, preferably, accepted as a true and accurate record, are better because they fall under the business records exemption to the hearsay rule if that is the normal practice of the organisation. Even better if the participants sign them.
Similarly, writing the notes and circulating them, by email for example, improves their veracity because it creates a verifiable timestamp and gives other parties the opportunity to disagree with the author's version while the memories are still fresh.