Now since the example is "I" being the subject, so there're no difference of sentence structure using either 自信 (self-assurance, confidence in yourself) or 信心 (confidence in yourself or someone/something else). The difference will occur in a sentence like this:
Pinyin: rén men duì jīn nián de jīng jì qián jǐng méi duō dà xìn xīn。
Meaning: People don't give much hope to the economic growth.
Note: In this sentence, don't replace 信心 with 自信, though I can't say it's utterly wrong to use 自信 here.
In your example, you can use both 信心 and 自信, because "confidence in oneself" is the common point between these two words:
Pinyin: zài wǔ tái shàng tiào wǔ，wǒ méi xìn xīn。
Pinyin: zài wǔ tái shàng tiào wǔ，wǒ méi zì xìn。