After looking into the grammar to be sure, I can say with confidence that the edits are correct.
The て-form can behave in several different ways, and as you have noticed there are times where it behaves more like an 'and' in some sentences, but this is not the only function. There are many grammars that use the て-form, like ～て + も, ～て + いる, and others.
In this particular case, the て-form + 過ごす is another form of grammar meaning 'to spend time verbing'.
Here's a webpage with examples of how to use 過ごす. Sentences in which 過ごす modifies a verb uses the て-form, and I will list only two of the many examples from the webpage here:
I often spend my leisure time listening to the radio.
I will spend next Sunday reading novels.
Grammatically, the construction is simple put a verb into the base-て form, and then add 過ごす. Then, you get the phrase 'pass time verbing.' The amount of time being passed depends on the context of the rest of the sentence, and it needs to be designated. However, the general construction really is this simple.
So what's the difference?
I'm sorry to say it, but your initial attempt was grammatically incorrect. So sadly, the difference is simply what is and what is not grammatical. I'll copy the pertinent part below and break it down for you.
As you are no doubt aware, the を particle denotes a direct object. 過ごす, as it turns out, takes a period of time as the direct object. As 日本語を勉強すること does not mark a period of time, it is grammatically incorrect. The period of time does not have to be specific.