"I am at home too."
Translation:Ja też jestem w domu.
So it's like:
'Ja też jestem w domu' = 'I, too, am at home'
'Ja jestem też w domu' = 'I am at home, too (along with being in another place)'
Because that sounds as if you have the gift of bilocation - you're in the house and in another place as well.
But the English sentence "I am at home too" also can imply that you are in two places at once (one of it being a home). For me both sentences are corrent. You can also say: "Też jestem w domu".
To some extent the ability to judge the intended meaning is important. Sure, there are numerous sentences that might be understood in more than one way, sometimes contradictory („Motocykl wyprzedził autobus” anyone?).
What is, without a specified context, more probable? That you are at two places at once, or that you are at a similar place to the one another person is? Each one can try answering it oneself, but „Jestem też w domu” leans in meaning towards the former
"Też jestem w domy." is considered incorrect. Why is that? I thought we could leave out the personal pronouns...?
Apparently, the typo ('domy' vs 'domu') was enough for the system to reject your answer. "Też jestem w domu" is accepted.