Detect if word is «common English» word or slang word
It all depends on your definition of what a common word is in your domain. You are using an NLTK corpus which likely doesn't fit your domain very well. Either you have a corpus containing the domain you want and you do a simple lookup. Or you don't know in advance and you need to compute these common words from your documents (your short phrases). In that case, the more sentences you have the better.

An easy way to do this is using pure Python is to use a counter

from collections import Counter

documents = [] # here add your list of documents/phrases

counter = Counter()

for doc in documents:

    words = doc.split() # assuming that words can be split on whitespaces

    counter.update(words)

counter.most_common() # this will return words ranked by their frequency


Then it's up to you to apply a threshold to define what words are common and which aren't. A more advanced approach could be using their TFIDF weights but in that case, you are not necessarily keeping common words, you are keeping "important" ones.