if initially the air in the house is humid, the system dehumidifies the air, and then once the RH% has dropped into the 50s, it concentrates on actually cooling the air ?
I am not an HVAC pro even though I used to mess with big chillers but your restatement is correct. One way to think of it is the air in the room is a mix of two distinct parts. Dry air and water vapor. The inside coil in the AC unit will be colder than the room temp , water vapor in the air will condense on the coil and down into the drip pan and hopefully outdoors as long as the vapor in the air is above its dewpoint. Once the air is below dewpoint then the amount of energy used is far less as the latent (hidden) energy being used to condense the water is far less. This latent load sucks up a lot of energy 970 btus/lb of water in the air. In order to know how much water in the air you need some basic room measurement tools and psychrometric table, there are lots of references on the internet but here is one
https://extension.psu.edu/psychrometric-chart-use. It looks intimidating but it tells you a lot, once you figure it out a you can establish the exact worse case room air condition and also the desired room condition and then calculate directly the cooling and dehimidfication load. Dry air is very light so it takes far less energy to cool the air but its enthalpy changes with temperature. Manual J estimates the sensible heat load coming into or leaving the room but does not deal with infiltration which can vary quite a bit depending on construction. A blower door test can measure the infiltration very accurately. Most people guess and usually guess wrong by guessing too low.
When you first turn on the unit, its going to be running full bore to drop down the humidity in the space and then it will start cooling the air. Ideally the unit is run early to knock down the humidity and then run on low to keep the space cool and deal with any humidity that leaks in or is introduced into the space. Folk like my deceased parents only had AC late in life up north and they always waited until the conditions in the house were near intolerable before turning the AC on and then my mom would complain that she didnt like it as it was "too noisy" It was noisy as it was struggling to deal with all the moisture. If I convinced them to run it earlier in the day, it would make less noise and the room never got uncomfortable. You are not just dealing with the humidity in the air, all the "stuff" in room that is permeable will have absorbed moisture and it needs to dry out so as the room humidity drops, this extra moisture enters the room and needs to be dealt with. Dealing with a bathroom with tub and showers can be a real PITA, in many cases its better to exhaust the damp air outdoors and let infiltration bring in outdoor air to replace it.
Note that if the space is too "tight" the air can get stale inside the room and some folks will run with a window open. Generally, if its a designed system there will fresh air inlet built into the system but usually a window shaker type unit does not and depends on air leaks in the room.
You will see warnings on occasion not to oversize an AC unit. It is somewhat rare for homes to be an issue but in commercial and industrial it can be an issue if there is lot of humidity in the room. The temperature of the indoor coil can drop below freezing and any water condensing on it can freeze and plug air flow through the coil rather than flowing to the drip pan. It is called "freezing" the coil. If a house is freezing a coil, its usually a sign that it is a very "leaky" home in a very humid area or something inside of conditioned space like cooking is adding lot or moisture. Single speed units tend to freeze up the most, two speed units are better and inverter units probably the least. There can be internal controls to prevent frozen coils but that usually means slower moisture removal. An inverter style unit has a wide range of cooling while a two speed unit only has two.
There also is the "dirty secret" of cooling in the past and that is reheat. Hospitals and public buildings want a lot of fresh outdoor to be circulated through the building to reduce odors and to keep nasties from recirculating the building. That fresh air is loaded with humidity that needs to be removed as they also want dry air as nasties in the air and on surfaces don't like dry air and they don't want any spots where mold might grow. That means there is a lot of latent cooling load dealing with all the humidity coming in. The approach used to deal with this was to overcool the air coming in and then reheating it with steam and electric coils to bring the air back up to comfortable. It works great but not very energy efficient. This was standard in most big buildings until energy efficiency came in and then buildings got tightened up and air was recirculated. This on occasion lead to "sick building syndrome". It is an issue with new super energy efficient homes like Passive Homes and the solution is a separate fresh air make up system that pulls air out of the building and runs it through a heat exchanger to pump back into the building. The heat exchangers used have two types, for a cold climate they use a latent and sensible exchanger to grab any moisture that may be in the indoor air being exhausted and put it back in the house while in cooling dominant climate, they use a sensible exchanger that allows the moisture to go outdoors while the sensible cooling is used to precool the incoming air.
In some cases, older residential system had electric reheat coils. Note this is not the same as strip heat which usually kicks in when outdoor temps are low and a single speed refrigerant system cant deal with the low refrigerant temps. Most folks do not know the difference and most residential techs are either mechanics who keep the refrigerant system working outside or they are duct guys. Few are really good with both skills and many systems are built to "rule of thumb". To actually design a system from scratch takes some technical background that most techs and small contractors skip. Few residential systems can afford to be designed by an engineer so in many cases the systems installed are just a guess based on similar systems.
