The particle 着 indicates the continuous aspect, and if we want to use this particle, the appropriate structure is:
Place + Verb + 着 + [Noun Phrase]
Alternative existential sentences; Pattern with 着, Chinese Grammar Wiki
So the "place" should come before the verb (as usual), i.e.:
It's continuous, so it is happening and will continue to happen. The context will give the time period in which it occurs. It means what you said: "Many students are sitting on the grass." (Although, I feel 同学 is better translated to "classmates" than "students", but this depends on context.)
We generally think of 坐 as intransitive, i.e., it doesn't take an object (i.e., "I sit" 我坐着) For comparison, 吃 can be transitive (e.g., "I eat food" 我吃着饭). Although we can say:
很多同学坐着地铁。 (Many classmates are taking the subway.)
but here 坐着 means "take", and 地铁 is the object of the sentence.
The official version:
uses a different grammar structure with the "special verb" 坐:
Subj. + [Special Verb] + 在 + Location
Special cases of "zai" following verbs, Chinese Grammar Wiki
And I believe your translation is correct: "Many students sit on the grass."