In China,those three ways all make sense and can be understood wherever you are.But,there is a little difference: In mainland China,“出租车” is common. If you want to take a taxi, you can say "打出租车" or "打车". In Hongkong and Macow and Guangdong Province, "的士"(pinyin:di shi) is common. Taking a taxi can be said "打的" In Taiwai,"计程车" is common.Taking a taxi can be said "叫计程车". In a word, they are all correct, just different usage in different region.

“出租汽车” is used no commonly.“计程车”is often used in Taiwan.“出租车” in mainland China.“的士”in Hong Kong,but also sometimes in mainland China(in oral).

Taxi" in Chinese: 出租汽车（or 出租车， in Mainland China)，计程车 ( in Taiwan, China)，的士( in Hongkong, China, and occasionally in Guangzhou/Canton and some other cities in Mainland China)

出租车

I think there is no difference among them.They are named by different areas in China.