1. 坚持;坚决认为(或主张、要求)
    If you insist that something should be done, you say so very firmly and refuse to give in about it. If you insist on something, you say firmly that it must be done or provided.

    e.g. My family insisted that I should not give in, but stay and fight...
           家里人坚持认为我不应让步,而应该留下来继续抗争。
    e.g. She insisted on being present at all the interviews...
           她坚持所有采访自己都要在场。

2. 坚称;坚持说
    If you insist that something is the case, you say so very firmly and refuse to say otherwise, even though other people do not believe you.

    e.g. The president insisted that he was acting out of compassion, not opportunism...
           总统坚持说他如此做是出于同情而不是为了投机。
    e.g. 'It's not that difficult,' she insists...
           “并没有那么困难,”她坚持说。