一杯: one cup and a state of being full
Compare the structures お茶を一杯 and 人で一杯 without treating one drink as a specified volume or proof of a full container.
Counting one drink
一杯, read いっぱい, can count one cup of a drink. 朝に温かいお茶を一杯飲んだ。 says that one cup of warm tea was drunk in the morning.
一杯 gives the drink quantity associated with 飲んだ. It does not specify the cup size, milliliters, or whether the cup was filled to capacity.
A space full of people
会場は人で一杯になった。 says that the venue became full of people. Here 一杯 describes a state rather than a cup of people being drunk.
人で identifies what fills the space, and 一杯になった reports a change of state. Read the venue together with the predicate structure.
A quantity and a full state
お茶を一杯注文しました。 says that one cup of tea was ordered. コップが水で一杯です。 describes a glass full of water.
Check whether the expression gives a quantity for an action or describes something filled with something else. Do not assign a fixed volume to every 一杯.
Does お茶を一杯飲んだ。 establish that the cup was filled to capacity?
Show the answer and why
No. It reports one cup of tea consumed. Cup size and how full it was require separate information.