How should I exit/close Pokemon Go?

You're not closing it properly unless you swipe it away. That's the app developer's fault (or by design as far as we are concerned) Pressing the home button only minimizes and pauses the game, swiping it, actually closes it, and when you re-start it it is forced to actually call the server, load properly and give you the latest Pokemon in your area. Sometimes when there hasn't been anything near me, I swipe it to reset the Pokemon around me and reload properly ...

It depends on which phone you're using. If you are using an iPhone, you will need to "swipe" it to close it.

If you are using an Android OS phone (as it appears you are from the image provided), you can still "swipe to close", which will safely close the app.

However, in extreme cases, you can "Force Stop" the app from the app settings menu. However this immediately and forcibly closes the app, killing all processes that may shut down the app safely. Doing so may cause a loss of data.

With the latest version on Android you can now close the app using the back button. You will get a confirmation dialog to do it like shown on this image: