ALL electrically grounded equipment MUST be bonded "connected together" Lightning rods are intended to slowly discharge static buildups through corona discharge so that the voltage never becomes great enough for lightning to "strike". Connecting them all is really the right way.

By 'arrestor cable' you mean the down line for a lightning rod?
If it takes a hit (they can and do) the lightning is very unlikely to make a turn into the house if it is near the earth.
When the high current present comes to a bend in the down line magnetic field crowding on the inside of the bend tends to force the lightning to travel in a straight line.
This is also the reason that all turns in a down line for a lightning rod should have as large a radius as possible.

All but the largest of down lines (think over an inch of copper) will need to be replaced if a direct strike occurs.

Brick, you are correct. It comes down from the roof to a "T"
straight down into the ground and also makes a 90 degree turn into the basement where it is attached to a cold water pipe.
Why should the cold water pipe grounded? Isn't it already grounded from where it enters the house? Should it have 2 grounds?
I have 1/2 inch copper lines. I can assume they have never had a strike?

The cold water pipe is part of the grounding electrode system.
It (along with at least one other 'made' electrode) have been required for a while by the NEC.
Older installations may use just the water pipe.
All the grounding electrodes are bonded together to make sure there is no voltage between them.
This is a case of 'more is better'.
The earth is a rather poor conductor and every electrode lowers the impedance of the ground path.
The sharp turn will help to prevent an actual high current surge from following the line.
A 1/2 inch copper line will handle only the lower end of lightning strikes (the grounding and bonding manual is at work).
Lightning rods do provide both a bleed point to decrease the earth-cloud charge and a safer path if a strike occurs.

Won't this connection to the water pipe ensure that the plumbing will be electified if the lightning rods get hit?

Yes, but if everything is at the same voltage it does not matter.

I don't understand. If the plumbing was not physically connected to the lightning system, and it (the lightning system)took a hit by lightning, and I was in the shower, would I be electified?
Wouldn't it be better to have two separate grounds? One for plumbing and one for lightning rods?

You have been given the very best answers available anywhere. At some point, you must choose to accept them by faith or to reject them. Some independent, personal, study of electrical theory is in order. That is your responsibility. No electric current flows between things that are at the same voltage- look at the birds sitting on ONE line of the POCO wires. The Earth has electrical resistance. Two (or more) separate grounding points will have resistance between them and when current is flowing, there will be a voltage difference between those points- BAD BAD BAD. Bond all grounds together.

As long as the impedance between two points is low for the currents present, no voltage difference can exist.

All grounding electrodes for a single structure should be tied together.
This prevent ANY voltage difference from occurring.
Electricity does not travel wily nilly.
While a rise in the earth voltage is very likely to occur with the current from a lightning strike going to ground, it will not result in any significant current flowing back into the plumbing in the house.
The current is trying to reach the earth from a lightning strike, and the plumbing inside the house is not a path there.
A lot of things happen every day that are never noticed.
When a charged cloud passes over your house, the voltage of everything connected to the earth rises in voltage.
The electrical wiring, the plumbing, the DWV system if it is metal.
This is easily demonstrated by driving a ground stake a hundred yards away and running a wire to it.
Drive another stake at the end of the wire.
You can measure voltage between the wire and a local stake easily and fairly often.
The voltage can get very large if a charged cloud passes over (dangerously large).
You never detect this since everything in a single location normally moves up and down in voltage together.

Thank you for telling me what my responsibility is. I appreciate that. I usually don't follow information blindly, I try to figure it out. If I am annoying you, you have my permission to ignore me.
Brick, thanks for the follow up!