Not sure if you have the same Setting for your region, but in your Settings for the Phone app, do you have the option for Call Blocking & Identification?
For me, when I tap on it there is a setting for Business ID that can be toggled off. The description for the setting says it will display caller identification provided by T-Mobile, which is my cellular provider. You may also have other options to check there.
Don't know why changing the Show Contact Suggestions has any impact on the name seen in Caller ID. According to the description of that option, it is for Siri to suggest contact information based on emails and invitations sent to your Calendar. I have seen those suggestion come up asking me if I want to add the number to my Contacts, but do not see how it is associated with caller ID when receiving calls.
Siri also suggests new contacts based on your use of other apps, such as email you receive in Mail and invitations you receive in Calendar. (To turn this feature off, go to Settings > Apps > Contacts > Siri, then turn off Show Contact Suggestions.)
Based on the way you use Contacts, Siri also provides contact information suggestions in other apps. (To turn this feature off, go to Settings > Apps > Contacts > Siri, then turn off “Learn from this App.”)
Add and use contact information on iPhone - Apple Support (KG)