我觉得两种分析都算是“对”的。但更容易接受的分析应该是：将 every time 视为一个固定短语，将 every time I saw her 视为一种特殊的时间状语从句。关于这个问题，我查了一下有关资料：

▲《张道真英语语法》对此的解释是：……也有些时间状语从句不由连词引起：

Now that we have decorated the house, we can move in. 现在房子已装饰好，我们可以搬进去了。

Every time I listen to music I’ll think of it. 每当我听音乐，我就想起这事。

He didn’t tell me anything the last time I saw him. 上次我见到他时他什么也没告诉我。

He flinched each time she spoke to him. 每次她和他讲话，他都畏缩。

The moment I saw him I knew that there was no hope. 我一看到他，就知道没有希望了。

I came immediately you called. 你一来电话我就来了。

The machine will start instantly you press the button. 你一按电钮，机器就会开动。

Directly he uttered these words, there was a dead silence. 他一说这些话，大家立刻沉默下来。

▲《薄冰英语语法》对此的解释是：……有些表时间的短语亦可引导时间状语从句，如the instant, the minute, the moment, the day, the year, each (every) time, next time, the first (second, last) time, by the time等。例：

The minute he saw her he fell in love. 他对她一见倾心。

The day he returned home, his father was already dead. 他回家的那一天，他的父亲已经死了。

Every time I come, he is reading. 我每次来，他都在读书。

Next time you come, please bring your composition. 你下次来，请带来你的作文。

He was much better the last time I saw him. 我上次见他时他好多了。

By the time he was taken to the hospital he was nearly dead. 他被送到医院时已经几乎不行了。