Thank you.  I am refreshing my Sandbox now with the new fields.  I need this to capture all of the Tasks mentioned for any lead UNTIL it is in a "Qualified by PA" status and then, it should no longer count the activities.  I wil create it when the Sandbox is refreshed and keep you posted.

made some updates to this and also including the test class.  trigger passes at 100% in my dev environment with this.

trigger:

trigger taskTrigger on Task (after insert, after update, after delete) {
 if ((Trigger.isInsert || Trigger.isUpdate || Trigger.isDelete) && Trigger.isAfter) {    
 
   set<Id> LeadIds = new set<Id>();
  
  if(trigger.isInsert || trigger.isUpdate){
    for(Task p : trigger.new){
      LeadIds.add(p.WhoId);
    }
  }
 
   if(trigger.isDelete){
    for(Task p : trigger.old){
      LeadIds.add(p.WhoId);
    }
  }
   map<Id,Double> LeadMap = new map<Id,Double> ();
 
  for(AggregateResult q : [select WhoId,sum(Task_Name_Count__c)
    from Task where Status = 'Outgoing' 
                            AND 
                            (Type = 'Call' 
                            OR Type = 'Call: Reached'
                            OR Type = 'Appointment Kept'
                            OR Type = 'Appointment Scheduled'
                            OR Type = 'Voicemail left'
                            OR Type = 'Email')
                    
                    AND WhoId IN :LeadIds group by WhoId]){
      LeadMap.put((Id)q.get('WhoId'),(Double)q.get('expr0'));
  }
 
  List <Lead> LeadsToUpdate = new List <Lead> ();
 
  for(lead l : [Select Id, Task_Counter__c from Lead where Status = 'Qualified' AND Id IN :LeadIds]){
    Double Sum = LeadMap.get(l.Id);
    l.Task_Counter__c = Sum;
    LeadsToUpdate.add(l);
  }
 
  update LeadsToUpdate;  
  
 
 
  } 
}

 test class:

@isTest
private class TestLeadRollUpTasks {
 
    static testMethod void testRollup() {
 
        Profile pf = [Select Id from Profile where Name = 'System Administrator'];
 
        User u = new User();
        u.FirstName = 'Test';
        u.LastName = 'User';
        u.Email = 'testuser@test123456789.com';
        u.CompanyName = 'test.com';
        u.Title = 'Test User';
        u.Username = 'testuser@test123456789.com';
        u.Alias = 'testuser';
        u.CommunityNickname = 'Test User';
        u.TimeZoneSidKey = 'America/Mexico_City';
        u.LocaleSidKey = 'en_US';
        u.EmailEncodingKey = 'ISO-8859-1';
        u.ProfileId = pf.Id;
        u.LanguageLocaleKey = 'en_US';
        insert u;
 
        system.runAs(u){
 
            Lead l = new Lead();
            l.FirstName = 'Joe';
            l.LastName = 'Smith';
            l.Status = 'Qualified';
            l.Company = 'Test';
            insert l;
 
            system.assertEquals(l.Task_Counter__c, null);
 
            Task t1 = new Task();
            t1.WhoId = l.Id;
            t1.Subject = 'test';
            t1.Priority = 'Normal';
            t1.Status= 'Outgoing';
            t1.Type = 'Call';
            insert t1;
 
            Lead lu1 = [select Task_Counter__c from Lead where Id = :l.Id];
            system.assertEquals(lu1.Task_Counter__c,1);
 
            Task t2 = new Task();
            t2.WhoId = l.Id;
            t2.Subject = 'test';
            t2.Priority = 'Normal';
            t2.Status= 'Outgoing';
            t2.Type = 'Call';
            insert t2;
 
            AggregateResult ag1 = [select sum(Task_Name_Count__c) from Task where WhoId = :l.Id];
 
            Lead lu3 = [select Task_Counter__c from Lead where Id = :l.Id];
            system.assertEquals(lu3.Task_Counter__c,ag1.get('expr0'));
 
            delete t2;
 
            AggregateResult ag2 = [select sum(Task_Name_Count__c) from Task where WhoId = :l.Id];
 
            Lead lu4 = [select Task_Counter__c from Lead where Id = :l.Id];
            system.assertEquals(lu4.Task_Counter__c,ag2.get('expr0'));
 
        }
 
    }
 
}