You're logged in, except it's an error
Abstract it away!

Not sure exactly which instruction means to throw error 7 (which is normally an out of memory error code), but I think something like this should work - basically the idea is to extract the piece of logic into its own self-contained function, so that the calling code doesn't need to worry about an expected error and can use a simple Boolean value instead:

Private Function IsLoginEnabled(Optional ByRef outNoteElement As WebElement) As Boolean
    On Error Resume Next
        Set outNoteElement = this.Driver.FindElementByClass("note")
        IsLoginEnabled = outNoteElement Is Nothing And Err.Number = 7
        Err.Clear
    On Error GoTo 0
End Function


The calling code's logic can now leverage a simple Boolean result:

this.IsLoggedIn = IsLoginEnabled(noteElement)


It can also see that noteElement only ever contanis a valid object reference when the function returns False, so you don't need to be dealing with a run-time error in order to determine whether you need to inspect that noteElement object give your AP department a call.