Does their argument make any sense?
No. First, the [property manager's] offer & signature reflects that the renewal or extension meets the contract law requirement of being entered by the landlord knowingly and willfully.
Second, the landlord's allegation that "the reason why my lease is actually just $X is that it is a short term lease" is not credible. It is inconsistent with the common practice that shorter leases are proportionally more expensive than a full lease. That common practice is devised to compensate for the heightened overhead and risk a landlord incurs by searching for, vetting, and setting up new tenants as well as for the maintenance associated to tenants' turnover.
Third, if the August-December period is part of a lease shorter than yours, the rate of that other lease is relevant. That is because a rate which is materially higher than $X (again, proportional to the duration of the lease) would contradict the landlord's allegation that your lease is only $X on grounds that "it is a short term lease". Likewise, extending/renewing the lease at the same price is consistent with the landlord's allegation about short term leases.
Fourth, the Restatement (Second) of Contracts, which is frequently applied in the jurisdictions of the U.S., addresses at § 153 the issue of when a mistake of one party makes a contract voidable. The landlord can hardly prove that "[you] had reason to know of the mistake or [your] fault caused the mistake", Restatement at § 153(b). It was not your fault that the landlord offered/signed the renewal you accepted. Nor can it be said that you "knew" the offer of renewal was a mistake, since a landlord typically is interested in retaining its tenants.
Based on the Restatement § 158, you might be entitled to some arrangement or compensation in accordance with your next best alternative.
is there a chance that I would lose the case if I brought it to court?
Yes, but that depends on:
- judicial incompetence that has nothing to do with a true application of the law (see this judge's blatancy toward favoring "anybody who's powerful" and some ramifications of that mentality), or
- particularities of the situation which are not palpable in your question and which might trigger some narrow legislative provision.