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What is the practical difference between actions considered Pesik Reshe Delo Niha Le, Pesik Reshe Delo Ichpat Le, and Melecha She'en Sarich Legufa? What is an example that can demonstrate the difference between the three?
(Sources, of course [see the Rashash on Tosafot Ketuvot 6a])
As mentioned in the comments פסיק רישא דלא ניחא ליה is probably equivalent to פסיק רישא דלא איכפת ליה. The Big Question is regular פסיק רישא vs. מלאכה שאינה צריכה לגופה
Background
There is double machlokes that comes up between R.Shimon and R. Yehuda. R.Shimon holds "דבר שאינו מתכוון is muttar (Beitza 2:10), while R.Yehuda holds its assur. R. Shimon holds מלאכה שאינה צריכה לגופה is assur (rabbinically) while R.Yehuda holds it d'oraysa.
The Categories
דבר שאינו מתכוון - when you do one act, and another secondary action also takes place. E.g. if you drag a light bench to bring somewhere, and it might cause a furrow to be dug, (the melahca of plowing). The shulchan aruch et.al. hold this is muttar (like R.Shimon), even if the secondary act is beneficial.
A פסיק רישא is a davar sh'eino miskavein where the secondary act is for sure going to happen. E.g. if you drag a heavy bench somewhere, and it will definitely cause a furrow to be dug (which is also beneficial). This is forbidden even according to R. Shimon, and many views hold it is assur d'oraysa.
A פסיק רישא דלא ניחא ליה is simply a פסיק רישא where the secondary act is not beneficial (or worse). For example, if you get no benefit from the furrow your bench creates. While there were some who permitted this, most hold it is rabbinically forbidden (see שו"ע או"ח סימן שכ:יח).
The standard case of מלאכה שא"צ לגופה is if you blow out a candle to make the room dark, but not to get ashes (the purpose of the melacha in the mishkan). I.e. you do a melacha but not לגופה, for its "purpose". The Shulchan aruch et.al. again hold like R.Shimon, though some (the Rambam) hold like R.Yehudah.
Complexities
מלאכה שא"צ לגופה also comes up when you dig to get dirt, but it results in a hole being built (melacha of boneh). This leads to various complexities and there are many different interpretations on how it differs from פסיק רישא.
According to a standard explanation, when you do an action (digging) for a different purpose (getting dirt) we can show its in a different form than the melacha in the mishkan, but when you do a different action (dragging a bench), you're intent no longer matters, and it may be d'oraysa (of plowing). There are still some difficulties, so there are other explanations and divisions between d'oraysa and derabanan.
In Hilchos Shabbos 1:7 of the Mishnah Torah, The Kesef Mishna brings HaRav Avraham Hachasid (Rambam's son) and explains psik reisha doesn't need the actual melacha, he brings the case of closing a door on a deer.By melacha shaina tzricha it is the melacha you want but not its purpose, ayin sham.
I based this off one of the Aharonim I just don't remember right now which Aharon (I think it was the Rashash Ketuvot 6a).
By Melacha Sheen Sarich Legufa: the actual action is a Melacha even if you don't want to do the Melacha it is inevidble because the action itself is a Melacha.
By Pesik Reshe Delo Niha Le: The action itself isnt a Melacha but there is an inevidble Melacha that comes along with it.
Now this is based off a few other Mefarshim:
Pesik Reshe Delo Niha Le- is that he specifically doesn't want the Melacha that is occuring to occur because it's unncessary but it is inevidable
Pesik Reshe Delo Ichpat Le- he doesn't care if it happens but it is inevidable.
