I'm using Multibit. After sending transaction to chatbot my btc address came out to have 0 btc balance. I sent some btc to this address and overall wallet ballance has not changed (except minus transaction fee) and the linked address show now the newly sent btc. Should I now sent all my btc in Multibit wallet to this address? Is this something like sending bitcoin between different addresses in the same wallet? I'm a little wary not to make some fatal mistake.
Yes, you are sending bitcoin between different addresses in the same wallet.
As far as I can see from the part I've
bolded above, you are doing the right thing. Now just send all the BTC to the linked address, exactly as you did above.
BTW, Multibit should be showing all your addresses in the "Request" tab (though that depends on which version you are using). Make sure you see your linked address there in the list of your "Receiving addresses"!
P.S. BTW, if you are using Multibit, why don't you just use the signature method..? Its easy and free to use for as many addresses as you want