Coolant temp. raising. no coolant loss.

yes, might be, but you have to find the reason if that is not the issue. change and retest. check that your temp is correct and your fan (s) are functional and that the thermostat is functioning correctly too. As you say but there might be more to it.

Could be a defective radiator cap, or a leaking head gasket.

take it to a radiator shop and have the system pressure tested
if the pressure remains static at pressure start the engine and see if the pressure drops
if it does indicates a water pump seal when running
check the drain hole under the pump housing for stains and water marks
something the garage should have done

There are 2 tanks on a radiator. One on top and one at the bottom. They are what seal the top and bottom of the radiator core. You will also find an overflow tank connected by a small hose near the radiator cap that connects to the overflow tank. The overflow tank is where you top up the coolant. One some coolant system, the overflow tank is pressurised and the "radiator" pressure cap is on the overflow tank, on others the radiator pressure cap is on the radiator and the overflow tank is unpressurised.

Doesn't sound like it ran hot enough or long enough to crack or warp the head. If there is a crack, the engine will continue to overheat. If a warped head, car would probably run rough, the affected cylinders would likely have little or no compression, there would be a noticeable miss, and overheating would continue.
Why the low coolant? It's a sealed system, so should never have to add coolant, other than a small loss from evaporation in the overflow tank. Monitor the level closely for the next few days. Note the level in the overflow tank to see if it drops over time. Leaks can be external anywhere in the coolant system-engine, radiator, hoses, heater core- or internal in the engine-head gasket failures can cause coolant in the cylinders (white smoke out the exhaust), or coolant in the oil (oil will have a milky brown look to it), or oil in the coolant (will see traces of oil in the radiator).
A couple of tests may be called for. A pressure test of the coolant system will check if the system holds pressure as it should-about 15 psi. If pressure does not hold, there is a leak somewhere. Water pumps usually leak through the weep hole when they fail. A radiator leak would show up when pressure is applied. If pressure does hold, check if you have a good radiator cap-the cap is what seals the system so pressure will maintain. Coolant under pressure raises it's boiling point by about 10 degrees, so is critical to keep the system under pressure.
If you suspect further problems-losing coolant or continued overheating-then have the chemical test done to check for exhaust gasses in the coolant. Basically a check if the head gasket is good, a simple test done at the radiator cap opening with a special fluid that changes color if hydrocarbons are present in the cooling system.
Or you could have a compression test of the cylinders to check the internal mechanical condition-not only tells you if head gasket is good, also tells you a lot about engine condition-even,. balanced, good compression in all cylinders translates into good power output-rings are good, valves are sealing good, all systems go.
The cooling system: water pump, thermostat, radiator, hoses, heater core and hoses, radiator cap, overflow tank, and the cooling fans and associated wiring. Make sure everything is working right and you'll be good to go.

I am not sure that I understand your question or problem. All cooling systems are under pressure by design so that the coolant does not reach its boiling temperature. Since it is a closed system pressure should be equal throughout. What is happeneing?

you should have the cooling system pressure tested and inspected for a leak that may only show up under pressure.it could be a slow leak from the water pump,frost plug,head gasket which wouldn`t show up in the oil,heater core.it sounds like a small leak but this is the best way to start to find it.

Check for milky oil
It may help to have the radiator cleaned and flushed, and a new thermostat installed, the coolant may be going out the overflow
Not certain how it ruins radiator caps, they hold about 5psi in the radiatior and let excess pressure into the overflow, then when the engine cools, allow coolant to s uck back form the overflow tank

you need to replace the heater core and pressure text the radiator cap, replace if it leaks down go to a parts store to do this also you need to replace freeze plugs if you don't they will blow and ruin the engine,a slo check that the water pump is not bad, loosen the pump belt and see if you can wiggle the water pump with fan connected, if you have any play replace the water pump. and add new anti freeze.

Pressure test the system, it sounds like head gaskets that is backing into the water jackets and not the oil ports, which is the best of the two. Have heads checked for cracks of warping.