pedestal, a stand, counter for machines and vehicles
台 centers on a raised flat surface that supports something: a pedestal, stand, or platform. The counter sense extends this idea to machines and vehicles that sit on such a base.
台 is the modern simplified form of 臺, which originally depicted a raised platform or tower. The modern form 台 is a separate ancient character that was adopted as a simpler substitute.
The top 厶 looks like a private space, and the bottom 口 is a base. Imagine a private pedestal where you place a prized machine or vehicle.
For ダイ, picture a die-cast metal stand holding a machine: die -> ダイ, and the stand is the pedestal.
stand; rest; base; pedestal; platform; table; holder; support; rack
stage (of a theatre, concert hall, etc.)
Taiwan
typhoon; hurricane
Sendai (city in Miyagi)
script; libretto; scenario
cart (esp. a food cart); stall; stand
kitchen
spoiled; spoilt; ruined; wasted (chance, etc.); messed up; destroyed; (coming to) nothing
rise (e.g. of a movement); emergence; rearing one's head; gaining prominence; coming to the fore; gaining power; gathering strength
foundation; base; basis
lighthouse
platform truck; hand truck; trolley; dolly; cart
high ground; elevation; hill; plateau
offstage; backstage; behind the scenes
bed; couch
account book; ledger; register
(truck) load-carrying tray; (bicycle) luggage carrier; roof rack
meteorological observatory
astronomical observatory
(playground) slide
(major) mark; level; barrier
plateau; tableland; eminence
dresser; dressing table; vanity table