Which expression do native speakers actually use: Chinese New Year or Spring Festival (or other)?
The most common expression should be 春节.

农历新年 is a kind of rare.

We have another word for January 1st (not spring festival), 元旦(yuan2 dan4). So 新年 is normally the Chinese New Year.

Another expression is 阳历年(yang2 li4 nian2) for January 1st, 阴历年(yin1 li4 nian2) for spring festival. 阳历年 may be used more often, as 阴历年 is the default.