I'll try to explain. In your case you only have a 3 wire feeder to your barn. This is permissible under certain exceptions and is dependent on the NEC code cycle in force in your local jurisdiction. If your on 2005 or before and no local amendments to the contrary you may run a 3 wire feeder from the service equipment
to a detached building. This is only if you do not have any other metallic paths between the building with the service equipment and the building your feeder supplies. This means you cannot now or in the future run metal water pipes or phone or data cables or any other wiring or metal paths that use the grounding system of the dwelling and detached building.
This per 2011 NEC which allows you to have a 3 wire feeder that is existing.
In your case it was not existing
Now this is dependent on the code cycle your jurisdiction is enforcing as said earlier. Pay attention to the highlighted exception.
250.32 Buildings or Structures Supplied by a Feeder(s)
or Branch Circuit(s).
(A) Grounding Electrode. Building(s) or structure(s) supplied
by feeder(s) or branch circuit(s) shall have a grounding
electrode or grounding electrode system installed in
accordance with Part III of Article 250. The grounding
electrode conductor(s) shall be connected in accordance
with 250.32(B) or (C). Where there is no existing grounding
electrode, the grounding electrode(s) required in 250.50
shall be installed.
Exception: A grounding electrode shall not be required
where only a single branch circuit, including a multiwire
branch circuit, supplies the building or structure and the
branch circuit includes an equipment grounding conductor
for grounding the normally non–current-carrying metal
parts of equipment.
(B) Grounded Systems. For a grounded system at the
separate building or structure, an equipment grounding conductor
as described in 250.118 shall be run with the supply
conductors and be connected to the building or structure
disconnecting means and to the grounding electrode(s). The
equipment grounding conductor shall be used for grounding
or bonding of equipment, structures, or frames required
to be grounded or bonded. The equipment grounding
conductor shall be sized in accordance with 250.122.
Any installed grounded conductor shall not be connected
to the equipment grounding conductor or to the grounding
Exception: For existing premises wiring systems only, the
grounded conductor run with the supply to the building or
structure shall be permitted to be connected to the building
or structure disconnecting means and to the grounding
electrode(s) and shall be used for grounding or bonding of
equipment, structures, or frames required to be grounded
or bonded where all the requirements of (1), (2), and (3)
(1) An equipment grounding conductor is not run with the
supply to the building or structure.
(2) There are no continuous metallic paths bonded to the
grounding system in each building or structure involved.
(3) Ground-fault protection of equipment has not been installed
on the supply side of the feeder(s).
Where the grounded conductor is used for grounding in
accordance with the provision of this exception, the size of
the grounded conductor shall not be smaller than the larger
of either of the following:
(1) That required by 220.61
(2) That required by 250.122
Now forget the grounding electrode requirement .. you keep thinking this has something to do with equipment grounding .. this is not why you bond neutral and equipment grounding.
So if you are on 2005 or earlier you may, considering any local amendments to the NEC, run a 3 wire feeder from the Service Equipment
to the panel in the garage in accordance with the exception.
In your case you are not running your feeder from the service equipment but from a panel downstream of the service equipment which is required by code to be fed with four wires or three current carrying wires and metal conduit for the equipment ground..
Put a flow chart together ... your first panel after the service disconnect is fed by four wires which is correct but
the ground and neutral were not separated, major code violation, and you now have neutral current returning to the main disconnect panel over the bare equipment ground wire and
the feeder neutral. You never want normal system current on the equipment ground where it is not expected. This bonding is energizing all bonded metal, your equipment ground wires and feeder egc as you have given the neutral current other paths to use to get back to the service equipment and then to the source (transformer).
Your second panel is next to the other sub-panel it is fed by 3 wires (another code violation) and you bonded neutral and ground but you do not have your (case) main bonding means installed in the neutral bar. Any hot phase fault to ground or that metal will not trip a circuit breaker ... the metal will be just like a live bare wire and will shock or electrocute you if you touch it and a fault is present.
You ran a three wire feeder from the second panel to your barn (another code violation). In that panel you connected to the left bar with the feeder neutral and one branch circuit neutral ... on the right you connected your grounding electrode conductor and the branch circuit ground and installed the bonding jumper to the metal case.
The neutral is isolated from ground and the case in that panel which is incorrectly fed with a 3 wire feeder. This is telling me you think that the path to earth over the grounding electrode conductor is what is required to open (trip) your circuit breaker. As it is wired right now you will not be able to trip any circuit breaker on a ground fault.
So think about this ..without a complete low impedance/resistance circuit back to the transformer (source) for fault current over an 'effective ground fault path' your circuit breakers will never trip and thereby protect you from electrocution. Notice I said back to the transformer not earth .
As it stands you have no protection against ground fault and electrocution at the second sub-panel or the barn sub-panel.