As all the halides in the arctube, condense on the bottom part of the arctube, I wondered what would happen if I would rotate the arctube upside down, so that the location of the condensed halides would be moved upward, and the new cold spot would be deplated of halides.
It turned out that the lamp run-up much faster, and when the ballast reduced its current (The moment you see a sharp brightness change, which is usually occurs when this lamp is at mercury bluish color) the lamp was already at almost optimal brightness and color.
How this was happened?

The best I can think of is more surface area for the Halides to start getting evaporated.

The heat rises, so naturally if the halide pool (it remains there stuck by the surface tension even when it melts) becomes above the heat source, it will be the first thing to heat up and melt. Naturally some of the halides immediately condense back on the colder parts of the arctube, but that transfers there the heat as well (it acts as a heat pipe).
Normally the area above the arc is dry, so even if it get heated rather quickly way above the fill boiling point, it does not lead to the required fill evaporation, as the fill pool is in the coldest spot. When you put the pool to the top, the hottest part won't get that hot during the warmup, heating the rest of the arctube instead.

Now with just magnetic ballast there would be no sudden drop in arc current just because the arctube had reached the steady state temperature, as the current reduction is gradual during the whole warmup (and in fact the delivered power is at iot's maximum only at the full level).
Most electronic ballast reduce the current gradually as well - it indeed remains higher till further stage of the lamp warmup, but it starts to go down at the moment, when the delivered power reaches the desired power level and going further with the warmup, the current is regulated so the power remains constant. Even the car ballasts do not change the power that quickly (mainly to cause no visible changes).
So if there is any sudden drop in brightness, it is either a consequence of some cloud of some component reaching the arc).
Or sudden drop in the fill pressure when the pool above the arc just fully evaporates, so stops feeding the arctube with the gas. As the gas still condenses on the colder places, the net result is drop in pressure. Of course, when the colder places fully heat up as well, the pressure rises back, but that will be slower, so not that visible.