Hi Scottaskinosie,
You are partially correct here, the issue is each of the screens you mentioned are based off different tables. The CustomerList screen is based off the Contacts table, however the EditCustomer Screen is actually based off the Customer Table. That being said, both tables have a "Notes" field, but there is no link between them. The reason for this is, the Notes field for "Customer" would be general notes for that Customer as a whole, not for specific Contacts associated with this Customer. The notes field for the Contacts Table would be specific for each Contact.
There are several ways that you could set this up, but it would mostly depend on how you specifically will be using these fields. The easiest way I found would be to "add a linked field" in the Contacts Table, linking the Notes field from the Customer table to the Contacts table, then use this field to enter notes in and replace the note field with the one already setup in the Contacts Grid. This can be done by going to Customize->Tables/Fields, edit the Contacts table, and use the option "Add a Linked Field", where you will "Link Using" the Entity Table (which is associated with the Customer table) and the linked field will be "Notes" (remember the original Contacts note field is simply "Note", the one from the Customer/Entity table is actually plural "Notes"). Now you can edit your EditCustomer Screen, find the newly added Notes field and add it to your screen (you can likely replace the note field you added already with this one). Then, edit your CustomerList screen, edit the "Contacts" grid, and in Step 2of9 "Grid Columns", you could do a few different things here. The easiest would be delete the original "Note" field already in the grid, add a new field, and locate the one labelled "Notes". and "Insert Column".
Again, what I described above is the easiest way to get a note that you have entered from the EditCustomer Screen, to populate in the grid on the Customers and Contacts List Screen. You could do some more advanced work to have the Customer Note automatically populate whenever you add a note in the Contacts Table, or vice-versa. You could also technically have both fields show up in the Contacts Grid using "SQL Overrides". Please let me know if this helps you out, if you need further explanation we can add more, depending on what you are specifically looking to do.
-Ben