When we say hallel on days like Rosh chodesh, we don't say parts of two prakim of tehillim. This is often called "chatzi" hallel, or half hallel. In fact, according to http://www.5tjt.com/local-news/13099-our-better-halves the hallel we are left with is closer to 85% of the full hallel. Some siddurim refer to saying the "whole hallel" as "gomrim" we finish the hallel (as in http://www.daat.ac.il/daat/sidurim/mizrah/hol/rosh.htm) , and this makes more sense. The gemara in taanis 28b uses the phrase "gomer bahem et hahalel" and later refers to the shorter version as subject to "medalgin", skipping, but not half (though there might be other places in the gemara which use the word "half")
I understand that "chatzi" does not always mean an exact 50% but I'm wondering when the texts (either liturguical or halachic) began referring to the incomplete hallel as chatzi. Is there a talmudic phrase which would source this (and thus give it semantic value within the context of the gemara's language and use of the word "chatzi") or is it simply a linguistic shortcut introduced later?