What you have observed is not inconsistent RSI calculations. We have run numerous tests and can guarantee that the calculations are perfectly consistent.
I currently have two GBP/USD 30m charts running in two separate workspaces, using exactly the same indicators. I refreshed both charts a couple of hours ago, and both showed the same price and RSI at that time. There's now a discrepancy of nine points between the two. There's clearly a problem.
The reason why you see different results after reload, is that reload gets new data into your chart. Whenever you request the data, the reference point is the current moment and the request goes from the current moment and into the past - not the other way round. Thus, once the reference point has shifted the data will be updated and the study will have different input data and hence different calculation results. And this is exactly what happens when the reload is done.
Of course the RSI will be changed if the data it is based on has changed, and it's true that there are errors in nearly any arbitrary period due to the problems with your servers. However, it's possible to find periods of good data, and there are still inconsistencies.
When you say that refreshing the chart changes the direction in which the data is loaded, surely that shouldn't change the close price of any given bar, assuming the bar close occurred when your server was serving. And presumably the RSI calculation is still performed in the same way once the data is downloaded.