Could it be that the other accounts not get synced. Go to settings > email+accounts > tap the account and check if the contact box is check marked. And sync.
Please also create an account on WC so you can reply on this post.
Under settings > email+accounts > microsoft account there is no contact box only Email.
Email box is checked by the way. So I tried to add a new contact from the website outlook.com and the contact is visible on the phone.
So the sync between the phone and microsoft account works but I can not able to save contact from phone to outlook.
On my college's phone (same os version but another vendor) the save to option is visible when he tries to add a new contact
Only google account is visible. Adding a new contact to google account works but I can not select which account to be used:
According to a guide on windowsphone.com
a "save to" drop list should be between add photo and Name.
I think this is a glitch and factory reset is needed. Thank you for your help.
Hope you did not do the hard reset before checking this.
In people hub under people is also "showing".
Push on that and then you can checkmark the accounts you want to make visible. In your case I think only google is checked marked. Checkmark the other and than you can add a contact.