Is it uncommon to ask for my boss to re-write my job description to
accurately reflect the increasing umbrella of tasks now beneath me?
No. Generally that's the sort of thing that would be addressed in the context of an annual performance review. It would be reasonable to bring the matter up at your next review, and request that your position description be updated to accurately reflect the tasks that you're actually doing on a regular basis.
However, it may not make sense to immediately request a review or to specifically request that they rewrite your job description outside of the context of a performance review. It's possible that they've given you the new tasks on a once-off or trial basis, and that you may not continue with them longer-term. Your job description is there to reflect the things you're routinely expected to do, not necessarily everything that you may be asked to do.
So I'd say give it some time, and if your new responsibilities continue then definitely bring it up in the context of your next performance review.
There seem to be blurred lines here where I'm not sure if I was
promoted or just being taken advantage of.
Could be either, or it could be their way of evaluating whether or not to formally promote you (i.e. by seeing how well you handle the additional responsibilities, first). It doesn't sound like you've allowed enough time to decide anything either way.
Depending upon your locale, the responsibilities outlined in your formal job description may have a bearing upon such things as your minimum wage/entitlements (for instance, Australia works that way, however I think that's more the exception than the rule). Or your employer may implement its own policy of position-specific pay scales (regardless of locale).
In such cases, getting an employee to routinely work on tasks that aren't formally part of their job description can be a tactic employers use to reduce the compensation they pay. And I've had personal experience with at least one company that deliberately and methodically did exactly that, and on a very large scale.
Outside of such cases, however, your formal title/job description is not a significant concern (apart from looking better on a resume if you can put 'Senior X' instead of just 'X'). Having more accuracy is always better, but not necessarily very impactful in most tangible ways.
Generally try not to assume your employer is trying to take advantage of you unless you've got some corroborating evidence to prove it. Like if you've had the additional responsibilities for a year, performed well at them, but they flat out refuse to update your position/job-description or even perform an actual performance review. Otherwise you should at least start out by trusting them.
To reiterate; my suggestion is to just give it some time, and if the additional responsibilities continue then bring the matter up at the next "natural" opportunity. Try to avoid rushing/forcing the issue.