DevOps vs SRE vs Production Support Engineers
The best answer to DevOps vs SRE is here What's the Difference Between DevOps and SRE ? From the above series, you will understand Class SRE implements DevOps and both are working on a similar ground and it's pretty align goal for both of them. 

As a production support, you will be doing the task which you have mentioned in the question, apart, you need deep visibility, alert and notifications, latency, metrics, tracking, security, etc. lots of things to take care. And, it basically providing the service to the end users who is utilizing your application or system. Also, I see production support team have written the dockerfile and bash scripts to automate many tasks. So, whenever user faced any issue that first comes to Production Support team and it might went to other IT teams if it's not easily resolve. But, if you look at SRE, their sole purpose is to keep system in reliable state every time, maintain resiliency, etc. In some organization, the DevOps does this part too. 

From my perspective, all the three terms are same as their goal align and just differ in functionality. These functionality differs from organization to organizations. But, DevOps are intended to bring culture and shift in paradigm so that everyone will work collaboratively. Hope this helps.