4:00 AM
@Workaholic Good question, first, unlisted TeX command (those starts with a backslash) such as \limits and \displaystyle in Approach0 parser will be omitted by this rule: github.com/approach0/search-engine/blob/master/tex-parser/… . Thus will have no impact on search results.
\limits and \displaystyle are unlisted in our lexer rules (i.e. you cannot find them in lexer.template.l) because they have only presentation effect but no semantics effect. For a search engine, it is OK that these commands get simply omitted.
As for || and (), it is a more interesting story. Short answer is, \left( and (, \right) and ), as long as they are paired correctly, they are the same. Maybe source code is again a good explanation: github.com/approach0/search-engine/blob/master/tex-parser/…
By "paired correctly", I mean the "grammar" at least should be correct, e.g. \left(a\right( is wrong and \left(a\right) is correct.
But for | and \left| \right|, they are quite different in a sense that \left| and \right| is easy to pair, but || is considered difficult to "pair", because given a | symbol (without left or right TeX command ahead), parser does not know if it is a left pair or right pair.
An example would be, in math set language, {x | 0 < |x| < 2}, parser has no idea that the first vertical bar does not need to pair with another vertical bar, or at least parser needs some level of math knowledge to construct a "operator tree" for this math expression correctly.
It is possible, but not elegant to code these "knowledge" into parser. Therefore, currently Approach0 uses a ad-hoc method to deal with vertical bars: If they have \
If they have \left or \right\ ahead, they are considered as left/right absolute value pairs: github.com/approach0/search-engine/blob/master/tex-parser/…
If they are just single vertical bar, parser rather thinks them as "vert" token: github.com/approach0/search-engine/blob/master/tex-parser/…
Which introduces a problem, if you search \left| x -y\right|, you cannot get |x-y|. So I mentioned in my TODO that I plan to do query expansion that will interpret users' \left| x -y\right| query to \left| x -y\right| OR |x-y|.
@MartinSleziak has made a correct observation, these \left \right brackets except single vertical bars, should have the same search results.

11 hours later…
3:06 PM
@WeiZhong Thanks for clarifying things!