Yeah, nice source of confusion as well, tbh.
So Core (installation method) is using core 
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I remember believing that HACS itself was an add-on and therefore could not be used with a Container setup. It took me a while to grasp the difference between the different types of extensions of HA. I still don’t always understand why some things are in separate containers and others are integrations.
Addons are a way to use 3rd party software in HAOS. At their core, they are unrelated to HA, and are not even necessarily referenced in HA (examples: SSH addon, Tailscale, …)
Integrations are doing the interfacing between HA and 3rd party software.
More than often, if you use an addon, you will also use a corresponding integration so that HA can “talk” to the addon. E.g. MQTT.
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…best to avoid that word… 
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