V了₂ at the end says a new state now holds (it got cold); 了₁ clipped to the verb says an action was carried out (I drank it). One reports a changed situation, the other a finished act.
Both sit at the end and mark change. V了₂ leans on the verb or adjective in front of it to name the state reached; 了₂ is the same final particle read on the whole clause. Same 了, narrower spotlight here on the predicate's new value.
treating it as past tense: 昨天天氣冷了 (no change reported) → 天氣冷了 (it has turned cold)
negating the state: 天氣不冷了 keeps 了 for 'no longer cold'; ✗ 天氣沒冷了
clipping 了 to the verb when no object follows: ✗ 天氣變了天 → 天氣變了
English reads 了 as a past-tense ending and stamps it on every completed verb. V了₂ is not tense; it announces a state that has newly arrived.