Help Google contacts app

[Android Question]

I have a Samsung Galaxy S6 active Android version 6.01
I just entered a phone number for someone in my gmail contacts on my laptop. Then I went to the text messages app on my phone, opened a new message and typed in the contact's name, expecting the phone number to pop up - it did not. Her phone number does not show up show up on the phone, either, when I try typing in her first name on the key pad.
So it appears that something is not syncing. I have been working on this for an hour and a half trying to figure out why. The phone indicates that syncing of my email accounts HAS occurred. And I manually synced google just in case (although it had just synced).

I noticed that my contacts app on my phone is generic (as is the phone app) - generic or built in samsung - and saw on forums a reference to a Google Contacts app. I looked for Google Contacts app on phone and it is not there anywhere - and I mean anywhere. I searched on Google Play and there is no such animal.

So I am stuck. Can you help?

 

[svim]

Just to confirm in your Settings >> Accounts >> Google, when you tap on your email address in the listing of various services the 'Sync Contacts' box is check-marked, yes?
In your Contacts app, when you tap on the menu button there should be options for different things like 'Settings' and 'Accounts'. In Accounts, is your gmail address in the list? Is you texting app in the list? If not there should be an 'Add account' button so you can add them both.

Also, on your laptop go here:
https://security.google.com/settings/security/activity?pli=1
and you should see your phone in the list of devices linked to your Google account.