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We agree that the two situations are separate

Ok, that's a start.

but the first result is 4 ÷ a the second result is 4a

Exactly! So when a=2 then 4÷a=2, and 4a=8, which isn't the same thing. Welcome to why 2a and 2xa (and therefore also 2.a) aren't the same thing.

I use the dot as an expression of the same term rather than separate.

But that is incorrect. A dot is used for multiplication. i.e. it separates terms. If you use a . for 2.a, then you are writing the same thing as 2xa, not the same thing as 2a.

This is matter of my notational convention

Well, that's fine enough if you keep it to yourself, but don't use it in anything anyone else is going to read, or you're going to run into the issues I just pointed out