By Outlook do you mean the Windows program? If so then that is exactly how I handle my contacts - all entered into the Outlook app.
Scooter47's observation sent me into the Outlook contacts list to see how the phone numbers are formatted. What I found is that the problem is just as Scooter47 described. Contacts with phone numbers shown with the area code between parentheses come through with names to my Android phone just fine, however the names get lost when the phone is syncing back to the Phone Link app.
I tried reentering these problematic phone numbers as Scooter47 suggests, however Outlook proves to be pretty persistent in reformatting them with the area code between parentheses. I imagine there's a setting somewhere in the Windows jungle that controls this, but I am too lazy to search it out. Instead I found that Outlook does not reformat contact phone numbers if they are entered as a eleven digit string (1-country,3-area code, 7-number) with no separators (i.e. - ( ) +). Entered this way the names stay with the phone numbers down to the phone and back up to the Phone Link app.
Or at least it seems to work for me...
While being happy to see that the Phone Link app has become much better over the last year or two, it seems to me that Microsoft should have dealt with inconsistencies like this by now.
EDIT: Well, it worked for a while... Now it does not - at least not consistently. Cannot see any rhyme or reason why some contact names come across fine and others do not. Come on Microsoft! This cannot be a bug that is that hard to track down and fix!