IMHO, and in your case, it is. You already proved that it reads the same as your external SWR meter.
So Shadetree Mechanic and tba02 answered your question as to why you see a different SWR with the amp on or off. The input impedance of the amp changes when it is on or off. If you look inside your amp, you will probably notice wires (not coax) going to the SO-239 connectors, the wires certainly aren't 50 ohms impedance. They either go directly to the relay or to the circuit board where they attach to the relay.
In the off mode, your signal goes form the radio to the amp relay, flips over to the other side of the relay, and right back out of the amp and on towards the antenna. None of the path way in the amp is designed to provide 50 ohms impedance. The relay is designed to switch electricity (AC and/or DC) not RF. They make relays to switch RF but they are considerably more expensive and not usually found in a CB amp.
https://www.mouser.com/Electromechanical/Relays/High-Frequency-RF-Relays/_/N-5g33
When the amp is on, the signal goes from your radio, into the amp, into the relay, but from there takes a different path. It goes to a matching network and ultimately a splitter. This matching network has an impedance that is much closer to the output impedance of your radio, hence the lower SWR reading when the amp is on.
For an initial tune, your radio's SWR meter or an analyzer will work fine, but you should always do your final tune with your station setup the way you are going to run the majority of the time. If you are predominately going to run barefoot, connect the antenna to the radio and tune for lowest SWR. This configuration will give you the strongest TX & RX (coax and insertion loss goes both ways).
If you are going to be using your amp most of the time, then a couple of things. I would suggest using a low pass filter after the amp to keep the family and neighbors happy and a Common Mode Choke at the antenna feedpoint to keep you happy. The CMC will keep any RF coming from your unbalanced antenna from traveling back down the coax shield and, more importantly, any noise that has accumulated on your coax from getting into your antenna and conversely into your radios's receive. Depending on the series resistance of your CMC, you can lower your noise by 25~30 dB. Which means it will be much easier to dig out those weak signals. I too would run a SWR meter as the last thing in line if you are going to run an amp. This way you can keep an eye on things. Just tune the antenna for the lowest SWR on your external meter.
Just my opinion, YMMV.
Good Luck!
