And that is what I get for posting in the middle of the night while in bed.
I know they are homonyms and not synonyms. *sigh* Oh well, I suppose it happens to all of us sometimes.
Anyway, to return to the actual topic and not linger on my 2AM mistake...
I rarely hear anyone use わたくし立 in conversation, and have never heard いち立 from what I can recall.
The most distinction people around here give it is occasionally using 私立 vs 市の. But that doesn't help when the actual name of the place is 市立-something something.
I think this problem is particularly prevalent in my area as it fairly recently became a 市. All the formerly 町立 facilities have just had their names changed over to 市立.