There sure are a lot of different names for those things.
Embroidering on beanie/knit cap/toque brims
Ravelry: Folded Brim Beanie pattern by April Diodato
CVTVLIST SB I Embroidery Beanie
CVTVLIST SB I Embroidery Beanie - ニットキャップ/ビーニー
Wholesale Cashmere Beanie with Pompom Custom Embroidery Beanie
CVTVLIST SB I Embroidery Beanie - ニットキャップ/ビーニー
Farah logo beanie hat in black
How to Embroidery on Knit Hats & Beanies
Material:Cotton, Department Name:Adult, Origin:Mainland China, CN:Zhejiang, Applicable Scene:Casual, Gender:WOMEN, Feature:Keep warm, Applicable
PIKADINGNIS New Knitted Beanies Hats Cartoon Cute Smiling Face Embroidery Cap for Winter Women Cap Warm Melon Cap Street Head Cap Women Hat
Customizable Port & Company® Knit Beanie Hat
Charming Cat Hat Knitting Pattern - Little Red Window
Hello Cozy Hat - I Like Crochet