I think scheme should be like this:
You should subscribe on firebase updates somehow (I'm not familiar with firebase but I'm sure that there should be some mechanism for that) and when updates come from firebase - you should mutate your store by dispatching some actions, to keep you store in sync with firebase.
Ideally, your app should work with your store only. Store should hide such implementation details as firebase usage. If tomorrow you will switch from firebase to something else, you shouldn't rewrite your components which work with the store. They still will work with the store and with the same actions.
When you update something in your store you need to update firebase as well. I think you can do it right in your actions, it should be fine. In that way, you can be sure that all changes which will appear in your store, will be in firebase. E.g. you call someAction
which actually do some firebase manipulation. If you subscribed on firebase changes you won't even do something else in that action, except sending some command for firebase, because your subscription will update your store. Or you can mutate the store immediately (aka optimistic update), then wait for response from firebase, and then decide leave that update or call some mutation to rollback (however here you should be very careful because of subscription on firebase changes).
Or, alternatively, you can write some plugin for your store, which will send updates to firebase only when some change in your store has been done. In that case, updates from your app firstly will appear locally in your store, then your plugin will send them to firebase (maybe such plugin even already exist, idk).
I'm not an expert in firebase, but tried to share some thoughts about how it should work in general. Hope this helps.
Oh, back to your question:
dbOrdersRef.push(order) or with a call to the store with for example this.$store.dispatch('setOrder', order) which don't updates Firebase
I think you shouldn't use firebase in components at all and should use your store instead. So I'd rather advice to use the second option. But you should implement firebase updates by your own, there are no magic updates from store to firebase by default (at least if you aren't using some plugin with that magic).