3 questions about kans

[or2az]

From Barts guide:
A Daiminkan is completed by calling kan on an opponent's discard when you are holding a concealed pung of the same tile. The set is displayed with the 1st, 2nd, or 4th tile turned perpendicularly if the discard was taken respectively from the player to the left, opposite or to the right.

A Shōminkan is formed by adding a self-drawn tile to a previously melded pung and "upgrading" it to a Kong. The stolen tile is placed on top of the sideways tile and parallel to it.

What do you call a second kan that is formed from the supplemental tile you drew (after making the first kan) and adding it to a concealed pung in your hand?
And how do you arrange it on the table? (I think two sideways tiles are placed in the middle, based on a recent game, but I'm not sure)
Lastly, in a daiminkan, the placement, why 2nd and not 3rd? I never noticed that before.

[Iapetus]

Whether you drew the tile from the live or dead wall doesn't matter. A kan called from 4 self-drawn tiles is an ankan, and a kan called from an open triplet and a self-drawn tile is a shominkan.

A better question would be: if you daiminkan, then ankan from the rinshan, how do the kan dora behave? In rules where the kan dora is opened immediately for closed kans, and after discard for open kans, presumably the 2nd kan dora would be opened before the 1st.

I cannot understand why one would specifically rotate the 2nd tile. The 3rd tile being rotated is a safer. Some people declare shominkan by placing the 4th tile vertically next to the called three, like in this picture. In that case, if the 2nd tile was rotated, a shominkan made from a pon called from left would be indistinguishable from a daiminkan called from opposite, creating confusion. With the 2nd tile rotated, a daiminkan called from opposite would be indistinguishable from a shominkan made from a pon from opposite - still bad, but at least the discarder stays the same, so furiten isn't broken. If you shominkan by placing the 4th tile horizontally, it doesn't matter if you rotate the 2nd or 3rd tile, but it's still better to choose the safe option.

[Gnom]

I never saw somebody do a shōminkan that way. It may be allowed in some places but normally you should put the 4th tile on top of the tile you previously called for your minkō. If the four are on the same line as in this picture then it should be a daiminkan, in this case from the person in front. I was not aware that it was recommended to turn the second tile instead of the third but it doesn't make for a big difference and I think most people randomly so one or the other...

[or2az]

I think I'm a bit more confused now than I was before. I will summarize and please correct me if I'm wrong.

In a daiminkan, for the opposite player, it doesn't matter if you turn the 2nd or the 3rd tile sideways.

The second kan that is formed from the self-drawn supplemental tile (after making the first kan) and added to a concealed pung is an Ankan and should be displayed with 2 tiles face up (the middle ones) and 2 tiles face down.

A Shōminkan is a self-drawn tile added to a previously melded pung and placed sideways on top of the other sideways tile.

If this is correct, why is the self-drawn tile in a Shōminkan referred to as a "stolen tile" in the definition?

Thanks for all the help, both of you. Have to go back and fix my rinshan kaihou post slightly.

[Barticle]

Wikipedia actually illustrates three ways of displaying a Daiminkan with the tile taken from the player opposite, the third being three vertical tiles and one horizontal above them.

If this is correct, why is the self-drawn tile in a Shōminkan referred to as a "stolen tile" in the definition?

Sleep deprivation, probably. Added to updates list. For now pretend it says "fourth" instead of "stolen".