Actually, the two examples you bring up are paragons of asinine design. I can't imagine it being easy to justify such design decisions, your attempt to do so was wholly unconvincing.
Perhaps exposing yourself to other languages might inform your responses on debatably the largest technical forum in the industry: one with many developers from widely varying backgrounds.
What is there to defend? Can you explain why it's "asinine design?" You don't actually elaborate as to why, and I've provided you both academic background in mathematics and prior art. I apologize if I've missed your point.
As for ~=, you need to keep in mind that Lua goes all the way back to 1991 and back then there wasn't a big pressure to make your syntax look like C (of Java or JavaScript) like happens today. Syntax used to be much more varied.
And, on the other hand, ~= looks too much like an attempt to spell out ≅ in ASCII, but the meaning is completely different.
If anything, /= (as used in Algol-68, Ada, Eiffel etc) makes a great deal more sense as an obvious rendering of ≠. It's unfortunate that C confused matters by reusing it as assignment.