update account custom field when lightning note is created

Please give a try with below code. It worked for me when I attached note to Account using "New Note" in both classic and lightnning and tested in my org. 

trigger updateAccountField on Note (after insert, after Update) {
    set<id> Accountids=new set<id>();
  list<Account> accUpdate =new list<Account>();
    for(Note n:Trigger.New){    
     // Gathering all the Account Ids to which notes are attached 
           if(String.valueOf(n.parentId).startsWith('001') && n.Title.Contains('Client Health')){
            Accountids.add(n.parentId);
           }   
        }
        if(Accountids.size()>0){
        accUpdate =[Select id,Most_recent_Note__c from Account where id in :Accountids];
        }
   // Comparing with note id and Account id and updating the Last_Note__c field wit note body
        for(Note nt : Trigger.new){
           for(integer i=0;i<accUpdate.size();i++)
            {
                if(nt.parentId==accUpdate.get(i).id)
                {
                accUpdate.get(i).Most_recent_Note__c= nt.LastModifiedDate;
                }
            }
      }
    if(!accUpdate.isEmpty()){
        try{
            update accUpdate;
        }catch(DmlException de ){
            System.debug(de);
        }
    } 
    }