You recommend using the Debug tab as you will be running the application locally on your device while having access to all of the logs, network traffic and more. You can also see logs using the script provided under the Test Tab.
You can debug installed app using Chrome, its complicated:
In the develop tap, open intelxdk.config.additions.xml file
You will see a line:
<preference name="debuggable" value="false" />
change this to
<preference name="debuggable" value="true" />
Build App using XDK and install on device.
Connect device to computer using USB.
Open Chrome and install "ADB plugin" extension.
Start the app on device.
On Chrome, click ADB icon and select "View Inspection Targets"
select the app id in the list displayed in Chrome, it will open debug console and u can debug and see console logs
Change the debug back option to false when u have to submit the app to google playstore