There was a Postscript section in my English Course-book, which simply teaches the colloquial English phrases as in "Give me a lift", "Let me have a look", "There is no point", etc. But "postscript" won't return any relevant results in Google, so I thought there might be a better expressive/suitable term for this phrases. Is there really?
The phrases are actually titled "social expressions", and that's what I would google for. I might throw in the word "common" for good measure, and search for "common social expressions".
The word postscript has nothing to do with the phrases at hand. It simply means "addendum to the book".
"Post script" is a phrase derived from Latin "post scriptum", meaning "after the writing", and is almost exclusively used to signal an addition to a letter after the closing salutation. It is also usually abbreviated "P.S."
The construction "PostScript" is a trademark for a software package used in publishing and formatting of documents. I think the appearance of this in the heading of your study book appendix is a result of the author/editor using the default construction of the page header and not proofreading carefully enough to notice that the trademark was not relevant to the subject matter.