I think it's important to regulate means of communication and availability between desktop and mobile device.
Problem: Google Chat is on my computer and my phone. Some people talk to me the same whether I am on my computer or whether I am on my phone. I cannot always digest their communication the same way across devices. This insufficient communication causes repeat information, makes it look like I'm ignoring them, and other wasteful behavior.
Potential Solution: Go offline when on my phone or do not use Google chat on my phone.
Problem: Sometimes the best way to communicate with someone for logistically critical information is through hangouts. (Data plan cheaper than texts.
SMS: It's nice to have it limited to 160 characters because it tries to limit the amount to digest. Though it doesn't stop some people with keyboards to send large amounts of information.
It does not matter to a lot of people whether the mobile device icon is showing on Google chat.
Best solution may be education; a digital communications class.