The quick (and unhelpful) answer is that either way works fine.
I think that everybody comes across this decision at one point, and the decision you make depends on the likely future of the website... which means it's prone to premature optimisation... but that's always the catch,
As you've probably guessed by now, "home" is in some ways a verb as well as a noun, thus why you're having trouble figuring out what to do.
The answer depends on a combination of interpretation of your website structure and how much time is available to you...
if you have very little time to work on this... then stuffing the 'home' action into another controller is often considered the expedient option. It works, it lets you move onto other (probably more productive) tasks.
However, I agree that sometimes it is good to step back and think about what you're doing and whether it could be done "better"...
in this case, though it's harder to define "better" - as it's unlikely that putting the home action in a new controller would be measurably faster... and if it's the only action int he controller... it's debatable whether it's better, architecturally, to just adding it onto an existing controller...
So we start in on what is mostly a philosophical debate... in other words, no answer will be "more correct" than the other- it's more a matter of taste and circumstance. In this case, the debate hinges on making the structure more RESTful.
To be faithful to RESTful architecture, you would indeed move the action into it's own controller... but you'd first have to identify what the entity is. The "home" page is often not readily identifiable as a specific db entity... it's more often a portal-page.
Sometimes you can pick an entity eg online shops will often have a home page that is actually a "products#index" variation, or sometimes the "home" page is a UserAccount#show page... but more often, your home page will not be simple, and will combine information from multiple entities... thus why it is difficult to decide what the "right" architecture will be.
If you cannot identify a specific entity, then there is a valid debate as to whether to move the action into a specific controller.
however, you can always create a new "entity" that is centred around the architecture of the site. This is especially likely if you are going to come up with other non-entity-specific pages for the site (eg T&Cs or an "about our company" page).
The usual fallback being a "PageController" (or similar name) which isn't linked to an Active Record model, but to a more nebulous entity, in this case a "page" which is recognisable to a user of the website (eg the "home page" and the "T&C page" and the "about page"). Each action would be for a specific page...
So, it's up to you as to whether this fits better with your view of the architecture of your system... and whether it's worth the effort... but that's my view on the debate. :)