They have different meaning in English. stay and keep. Generally example. 

If you keep something you have it. Example "that book I gave you to read you can keep it if you like."

Stay is to remain. " I am going to stay at a hotel tonight"

The second sentence is correct, and so is the explanation you provided (also known, well, simpler, as "past simple" :) )

The first, however, is incorrect. You can say:

The window stayed open. Subject (+article) + verb, second form + adjective. Past simple textbook case.

OR

The window was kept/held open. (disregarding for a second that "to stay" is inappropriate, as even in its transitive form it cannot be used that way) As mentioned, that's a past simple passive voice sentence. The subject, however, is still "The window", because had you exchanged it for the first-person singular pronoun, you would have said "I was kept quiet" and not "me was kept quiet".

Hi Elena,

In the case you mentioned I think

stay is like remained

ex: The window stayed (remained )open all day and the rain entered.

The store stays open all night

The man stayed with his dog during the operation

She stayed out all night

She kept me convinced

kept is liked continued

She continued convincing me.

She kept selling me ice cream

She kept saying she was on a diet

She kept being helpful

Hope that helps.

The window was stayed open.

Its Past indefinite Passive voice sentence Because you don't have subject there. you only have object and was/were and 3rd form of verb......    which is something like that... Object + was/were + 3rd form of verb.... its the construction of making passive voice sentence....

Now the other one. She kept me convinced...

Its a past indefinite Active voice sentence. In this sentence you have Subject then 2nd form of verb and then Object....... Which is Subject + 2nd form of verb + Object..... The difference is these both sentences are of the same tense but one is Active voice sentence and the other one io passive sentence......