Approximation symbol: Is $\pi \approx 3.14\dots$ equivalent to $\pi \fallingdotseq 3.14\dots$?

Any mathematical notation is ok as long as it is common knowledge in your community. For instance, I believe I fully understand the meaning of the $\approx$ symbol. However, I haven't ever seen the second symbol you provided.

To be on the sure side you should provide a definition of any relation symbol you don't consider to be common knowledge. This may happen as a short remark ("..., where $\approx$ denotes ...") or maybe as a table of the used symbols in the front matter of your work. As with any definition in mathematics, there is no right or wrong in the symbol/notion/etc. you use, only proper or unsound definitions.

Also: When in doubt, use the symbol that is used more commonly in the standard textbooks of your field. There is no benefit in being avant-garde at notation.

While it is certainly true that with the proper definition there is now 'wrong' notation, perhaps it should be mentioned that some notation is more suggestive and/or easier to work with than others, e.g. Arabic numeral vs. Roman numerals, the various symbols for the derivative, and countless others. The actual symbols are arbitrary, but good notation can certainly promote the flow of ideas more easily.

Also, do I remember correctly that Feynman gave up trying to invent more efficient notation for simple math when he was quite young because nobody could understand what he was doing?

A good notation has a subtlety and suggestiveness which at times make it almost seem like a live teacher. --Bertrand Russell