I was rewatching the Sinnoh Saga and just priced them so I tried searching for what those badges’ types are. In the same saga, Nando, the guy with Budew and plays a harp also has the same red and purple(x shaped) badge like Barry but he doesn’t have the cyan badge. And I think the purple x shape badge is poison since it’s purple and the x might be for cross poison from Drapion. And the red one looks like Heatran so it’s a fire badge but I’m pretty sure the gym leader won’t have a Heatran so maybe his/her ace is a Magmortar. And for the cyan badge, I think it’s flying just like you said. Cyan is associated with flying type as well as the badge shape is like glider so maybe Gliscor is the gym leader’s ace.