My entire life I have known speach to mean the verb that describes the act of speaking in the present tense, while speech to be the verb that is the past tense act of speaking professionally/publicly to a group.
According to google :The correct spelling is speech. “Speach” is an outdated spelling from hundreds of years ago and is now considered a spelling mistake.
Oh God, of course it’s Doc, I was very tired last night. I’m not a huge fan of speach but I kinda like OP’s distinction between public and private speaking. I can even imagine a mnemonic, the ee is the little faces of the audience looking up at Einstein/Doc speechifying, the ea are the two people looking at each other in a conversation.