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. Β 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.