I’m calling it—引っ張る is impossible to get without a vocab summary, which this particular vocab does not have. 
It seems to be an idiom with many meanings.
There’s a line in an early episode of Shirokuma Cafe where Shirokuma mentions a particular topic for seemingly the hundredth time, and Pengin says something like “you’re really not letting that go are you?” …I don’t have the Japanese sentence to hand, but it uses this idiom.
That would seem to align with the definition “to pull tight”.
Edit: found it!
引っ張るねぇ~
You’re not letting go of that, are you?
There is probably some room for interpretation in the translation here, but that’s what I see on this Anki card. I imagine the translation is lifted from DVD subtitles.
If you look up 張る on Jisho, it points out that as a suffix, it specifically means “to be prominently …; to be persistently …”
I think that’s a good way of thinking about it - 引っ張る is like 引く, but specifically emphasizing that the effort involved is large or over a long period. The way I heard it described that made sense was you would 引く something smaller than you and 引っ張る something bigger than you.
The last few example sentences seem really hard though!
I mean, you dont have to remember all the meaning at once right? And for a better learning of words with too many meanings like this, writing it down in a notebook and review it every day helps a lot.
Also, dont just remember a meaning alone, with each meanings, try to incorporate an object with it, it will make this word or any other words much easier