Recently Viewed comes with the little blue pushpin on all objects unless you change it to something else. Recently viewed cannot be used with inline editing. You can see that the little funnel on the right side. When it is darker grey, you can do the filters you need to see the data you need. I suggest changing the default view to "All" and then show all the records for that one. The inline editing tool is the little pencil.

The College/Student record (shown below) doesn't have record types, so you can choose another option and edit inline right away.

By the way, the two Westchester entries were included in your list views so you could use them for training on creating filters.

When an object has record types, you can only filter by one record type at a time. Below here's how to set up a record type filter.  Let's start with All Accounts. I've removed the filter on that one because this list view should show ALL accounts.

To Start a new List view,  go to List View Controls:

Click New or Clone.

You can set this list view's sharing settings. Sometimes I create List Views that only I can see because I'm working on something and don't want to clog up everybody else's visible lists. Play with this one!

Click the Funnel to see all the filters on this list view. We can see that the Account Record Type field is done.  Now you can add other filters-- by State, by Total Gifts, whatever field you want to see either empty or populated.

The pencil will allow you now to inline edit.  When you see the little pencil, that means you can add to this field.

I'll use the Enrolled List on the Contact Page to demonstrate.

Because there are record types here, we need to have that as a filter for inline editing to work.
Click Add Filter
in the Field List, Find Contact Record Type and select it.
Next Choose the record type you want to edit.

Click Done

Click Save

Fields with Picklist Values cannot always be edited. When they can't, the little lock shows up.
On this list, you can't change the Contact Name or Email, but you can change the Account name.

When you click on the pencil at the top, near the funnel, it will open the next editable field on the page.

If you want to edit more, click on the down-arrow beside the record and select edit.

Now you can edit the fields that were not "inline editable."