Skill Misconceptions

Nega Skills
An inherent skill of the Beast class, each skill is unique to a specific Beast and represents the beast rejecting humanity (Beast I rejecting humanity based on thinking they can only suffer until death, Beast II rejecting humanity due to thinking that they will only bring death to her and her children and Beast III-R rejecting humanity due thinking only itself is worthy of being saved). From what is shown in canon, Nega Skills therefore only focus on negating certain things and nothing else (Beast III negates Saver/Ruler skills and Beast II negates Noble Phantasm generated by the proper continuity of the History of Man.

Eternal Arms Mastery
A skill that allows a Servant to continue using their full fighting skills even when under mental interference. For a Servant to qualify for this skill, they must be famed for their unrivalled martial skill in their own legend. For example, Lancelot was famed for being the "Perfect Knight" whose fighting skills were enough to defeat an armed opponent with only a tree branch. Another example is Minamoto no Yorimitsu, who was famed for being the strongest mystery-killer in the Heian period. This skill is usually seen on Berserker class Servants since it counteracts the combat degradation due to Madness Enhancement. One thing to note is that their unrivalled fighting prowess has to be explicitly mentioned in-lore to qualify for this skill; it is not enough for a character to be highly skilled in gameplay since performance in gameplay and story is usually exaggerated to make a more enjoyable game and is not really shown during the story section.

Pioneer of the Stars
A skill that turns "All difficult voyages and challenges which are considered "impossible" turn into "events that can be realized"" aka making the impossible possible. From what we've seen in canon, this can only be given the Servants who are classed as "turning points" in humanity's history. The problem with this skill is that it is hard to determine which Servants' actions are great enough to be considered a "turning point" of humanity and which would just be classed as a Star-attributed Servant (one who has been recognized as a great hope in humanity's history). The best way I think to see if a Servant qualifies for the skill is to put history down as a timeline, with the Servant's life as a line splitting it into two, and seeing if the periods either side of it has had a fundamental shift in the lifestyle of humanity due to the Servant's actions.

Battle Continuation
A skill that allows a Servant to continue fighting even when taking lethal wounds. The common misconception when giving this skill is confusing it with the Endurance attribute: while high endurance requires a legend of being able to withstand high amounts of damage, battle continuation requires the servant to have a legend of continuing to fight even when they should have died by all accounts. Another common misconception is to the ranking of BC given, usually giving a higher rank that the feats given deserve. For a rank of A, the servant's ability to continue fighting would have to be in the same ballpark as the following legend:

The Cú Chulainn Standard for A-Rank Battle Continuation​
Ah, what's this? Wonder how you got me. Oh well! no u
Shit, I've been weakened for finally breaking too many geases and can barely stand. Better drag myself to a lake to drink some water while I bleed out.
Well, I have some strength now, best tie myself to this rock with my own intestines so I can throw all my strength into killing motherfuckers.
Three days later, the enemy is so terrified they refuse to even try to approach until they see a bird land on and start pecking at me. One of them finally comes up to check, unties me, my sword arm falls and kills him.
They still aren't sure, so one of them cuts off my head. Fuck that guy, let's see how he likes having bits of him lopped off.
Let's see you stab me without your good hand!

Double Summon vs Double Class
Both of these skills involve a single Servant having class skills from multiple different classes The main difference between them is how they were created. For Double Summon, the extra class skills were added as a result of external tampering during the Servant Summoning. For this particular method, classes skills may be combined but if none of the parts are a Knight class (Archer, Lancer or Saber). For Double Class, the extra class skills are added by taking two pre-existing servants incarnations and fusing them into one servant. From what we have seen from canon, this does not have restriction on Knight classes that the former method has. On the other hand, the other servant that has this skill was a joke servant with no explanation given for how the process occurred, meaning that it can't really be used on sheets until we get more information.

Instinct v EotM True v EotM False
At their core, all 3 of these skills involve the Servant being able to gain an advantage in a fight due analysis of the current situation. The confusion with these skills stems from two points: 1) When a Servant should have True or False for Eye of the Mind and 2) When a Servant should have Instinct or Eye of the Mind.

For the first point, both the True and False version have the same ability - the ability to avoid danger in combat and so increase the chance of winning an engagement. The difference comes from how the skill was obtained. The "True" version comes from a Servant who has gained a lot of experience though many battles and trials and so can use their intelligence and knowledge in battle to known when to avoid certain situations and increase your chances of winning. The "False" version however is gained not though experience but though an 6th sense. This means that the Servant was naturally able to avoid danger in battle, whether by high natural talent or a slight super-natural intuition. Both the "True" and "False" versions of the skill can improved though the gaining of experience so, as in life, if two servants face each other in battle with the exact same amount of experience, the one with the "False" version will have an advantage since they had natural danger avoidance on top of their gained experience.

For the second point, EotM and Instinct differ in terms of the precision of information given to the servant in the middle of battle. While EotM just detects danger in battle, Instinct gives the Servant a sense of the exact moves needed to come out the engagement victorious. In terms of an example: imagine there are two swordsman fighting and one is about to attack. They can either attack from the left, right or straight ahead. In this situation, attacking to the right will cause the servant to be countered and killed, attacking straight on will give a neutral advantage and continue the fight and attacking left will expose a critical weakness and win the fight. With EotM, the skill will recognize that attacking right is a death sentence and will nudge the Servant into attacking in one of the two other ways. However, a Servant with Instinct will not only be told that attacking right is a bad idea, but that attacking left is “the best personal course of action”. Therefore, the EotM skill is for people who are for fighters who gain an advantage though tactics and skill while Instinct is for fighters who always seem to know the best possible way to advance in an confrontation and quickly overpower the opponent by attacking their weak points as though they have precognition.