I just added google because it is not good to save password online
Store your personal information online is always bad. It includes some practices as you store it on your computer, and let your computer online or store your data on companies with Cloud storage services (Google, Dropbox, Drive, etc.) or powerful search engine like Google.
but google can not be blamed in this because they are even the one that helped me by notifying me, I did not have them any blame.
Sounds like the hacker used your gmail to reset password of your Twitter account.