The short answer is that "the work is done" always means "completed".
A point for consideration is what "the work" actually refers to. For example, any job can be broken down into tasks, and "the work" could refer to the entire job or just a task. Work can also be ongoing, so you could just mean your work for the day, and it will begin again tomorrow. ...
According to the Cambridge Dictionary
done adjective [after verb]
If something is done, or you are done with it, it is finished, or you
have finished doing, using it, etc.
If it's done then it's finished. If the work is still in progress, it's not done yet.
The following is an example of the past participle used in the present perfect, it is grammatically correct and is more common in the British English idiolect than in American English.
He's swum across the river
In normal speech, the auxiliary have is commonly contracted, so the he has becomes he's. However, in fast speech, it's easy to mistake ...
We use "swam" and "swum" as the past tense and the past participle respectively. It's archaic to use "swum" as the past tense.
So the second sentence (He swam across the river) is correct and idiomatic.