Short answer: No the accuracy of past forecasts is mixed with no clear variant being usually better
The most recent detailed analysis of how good the various projections have been was published in 2000: Beyond Six Billion: Forecasting the world's Population .
This table summarises the accuracy of past forecasts for population in the year 2000:
These errors are mostly quite modest, given that even the starting
population for each projection was not known with exactitude.
Despite this apparent accuracy, the fact that all these past projections
of the 2000 world population are too high is somewhat surprising. A
major reason is that fertility in several countries has declined more rapidly
than expected, and more rapidly than previous trends suggested,
over the past decade
The table and comments refer to the UN mid forecasts so don't completely answer the question.
The older UN forecasts are not readily available online, but the British Library keeps hard copies (see here). The numbers I report below are from a selection of those books.
First a chart comparing the current best estimates of world population to older UN forecasts:
It can be seen that many of the mid estimates actually underestimate the population until relatively recently where the overestimates get bigger (though the errors are not large and the forecasts overlap on the chart).
Here is a table of the underlying data:
The closest estimates to each actual are highlighted in red. It is only recently that the low forecasts seem to be the more accurate ones an mid forecasts seem to slightly overstate the actual population as the 2000 review suggested.
It is worth bearing in mind that population forecasting, despite being on of the most reliable sorts of forecast, doesn't have a good track record at the hard stuff. It is easy to project the number of 40 year-olds 20 years in the future (hint: count today's 20-year olds). The key things that are hard to predict are death rates (though they are a lot easier than they used to be) and birth rates (which have varied unpredictably for some time). A quick search of the archives of The Economist shows speculation about UK population collapse in the 1930s, population boom in the 1960s (with scary total population in the 70 millions), flat population in the 1980s and, most recently scary populations in the >70m area again (though driven as much by migration as birthrate).