The agents just use the coded term in front of them on the monitor. AA, just like other US carriers, will code any flight as delayed for WX if any aspect of the delay can be attributed to WX. This is with DOT's blessing.
That is different than the wording of the COC for purpose of covering hotel + meals. In this case the issue is whether the delay causing the misconnect and hotel/meal expense was within AA's control. That is a case-by-case analysis.
To your point, be firm and use facts. But, the agent is not lying and the place to deal with this is on the spot. DOT won't intervene if you assert that AA lied.
As to whether it's worth waiting in line for a voucher, that depends on your personal circumstances. If you can expense this, submit it to travel interruption insurance or simply chalk it up to the costs of travel, that is great. But, if you are on a tight budget, the alternative to 90 minutes in line, a bus to a dump and back may be a plastic seat at the airport.