Fantasy TTRPG reasons for climate change
Similar reasons to real climate changes. Just, with magical/fantastic effects triggering them.
Short Term: Particularly Large Volcanic Eruptions
Solar Eclipses (more so with magic that might extend the eclipse)
Wildfire Seasons (that are particularly harsh)
Meteor Strikes
Long Term: Massive deforestation
Massive industrial output (forges, chimneys, artificer labs, etc)
Also, much like your suggestions, a city-state or region of mages that constantly alter the weather for their own region might have odd effects either continent- or world-wide. (Mercedes Lackey tends to include magic that affects weather in this way in her series, including the Obsidian Trilogy and the Elvenbane series.) It doesn't need to include a heavy-handed chaos-theory fueled 'every small change affects everything' in the way David Eddings used it. Careless, negligent, or cruel mage-governed cultures can mess with the world intentionally or unintentionally. And, it would let a clever GM have the climate affected more regionally than globally if they want.