OK , I am going to talk about some of the code issues . Not so much as to explain why . This does not cover all cases , but I am attempting to cover what the OP is most likely seeing / working with .
For many / most residential customers , if you have an overhead service , the utility meter is on the outside of the house . Often on the exterior wall . I understand it is possible for the meter to be inside the house ( basement , maybe ) but it is not done that in my area .
If you have an underground service , the utility meter is normally on the outside of the house . Often in the alley , on a utility pedestal or a wooden pole .
The first disconnecting means / overload device down stream of the meter is the " main " / service entrance . This can be at / inside the " house panel " , or separate from the house panel . Or the house panel may be at some distance & separate from the " main " service entrance .
The service entrance conductors from the meter to the " main " / service entrance , consists of a neutral and 2 " hots " for the typical residential service .
If the " main " / is inside the " house panel " , the house panel is the service entrance . Code allows / requires , in this instance , the neutral bar and the earth ground bar to be electrically connected / bonded . And this is where the grounding electrode conductor normally terminates . ( The grounding electrode conductor may , by code , be terminated at the meter neutral bar , but less commonly . )
At that panel and only at that panel .
Any sub-panel must have separate neutrals and earth ground conductors feeding it and terminated at separate neutral bars and earth ground bars . And the 2 " hots " .
If the " house panel " is separate from the " main " ( for instance on the outside of the house and the house panel is at a distance , say the garage ) 2 hots , a neutral and an earth ground conductors must be run between the two . And the neutral bar and earth ground bar must be separate . Because , per code , the house panel is a sub panel .
Neutral conductors are terminated at the neutral bar and earth grounds and branch circuit equipment grounds are terminated at the earth ground bar .
Now , as to what I can see from your photo .
You have a circuit beaker , at which the " hot " service entrance conductors terminate . If there is no " main " outside / separate from this house panel , then that circuit breaker is the " main " and the house panel is the service entrance . And what I can see of your neutral and earth ground bars are correct / legal .
I can not see a grounding electrode conductor , so I can not address that issue .
If the main is separate / at a distance from the house panel , then the house panel is a sub panel and the bar at the bottom of the house panel should not be there , connecting the neutral bar and the earth ground bar . And your house panel is not installed legally / per code / correctly .
Now , is your " filter " working correctly . I do not know , neither do I possess meters / instruments to verify .
If my answers are to long winded or confusing , then you should not question what information you are given by qualified persons .
There is a reason a journeyman electrician must have on the job work experience and pass an exam . In my case , I had in excess of 4 years of On the Job Training and of class room training .
God bless
Wyr