Foreseeing the murder of Ann Lively in The Minority Report

PreCrime probably ignores the minority report because Burgess is operating the system directly at the time - years before he trains John Anderton as his protege. He's able to direct the arrest team to get John Doe, and follows that by murdering Ann himself. He murders her wearing the same style of clothes, and same method that was planned for murder #1, therefore the differences between the reports are very small (except for the capturing of Burgess's face, but even that is very brief compared to the overall length of the report).

The 'modern' use of the PreCrime video manipulation suite has two observers witnessing John as he uncovers the location of the soon to be committed murder. Its possible that the Ann Lively murder was before they started doing that, as they were developing the protocols around PreCrime, or that Burgess (knowing the planned events already) was able to keep everyone's attention away from scenes that did not help him.

Dr Hineman developed the technology, but Burgess as the cop designed how it was operated, and understands how he can use presence of echoes to cover the murder he commits after the arrest of John Doe. Once the original images are used to identify the location of murder #1 - he's trained the technicians to ignore what appears to be echo data:

BURGESS (nods) Yes... It's called an echo. The act of murder is such a violent disturbance in the future continuum that it sometimes repeats to the Precogs.

WITWER (beat, remembering) Precog Deja Vu...

BURGESS We teach the tech's to identify them and disregard...

However there is a wrinkle in his plan, because of the minority report of Agatha. It must have been a minority report when the murders are first reported by the pre-cogs. If they had all 3 reported murder #1 only - then he wouldn't have needed to 'lose' Agatha's data from the archive. Having the murder he commits entirely missing from the original reports would have been the perfect situation for him, however Agatha appears to have reported the second successful murder right from the start, so he needed to delete that data.

The males may have caught up with Agatha later but by then it is assumed they are echoes and can be ignored.

The fact that there was indeed a minority report needed to be hidden for two reasons:

1. It would have revealed Burgess's role in Ann's murder
2. It would have brought to wider attention discrepancies in the reports, undermining PreCrime which is his lifetime achievement.

So why does the movie treat Agatha's vision as a minority report and why does PreCrime ignore it?

At no point does the movie actually do so. You seem to be confusing two different concepts from the movie. The vision of Ann Lively's murder isn't classed as a minority report at all. The whole fuss about the title-giving "minority report" was about John Anderton's minority report and his hunt for it (only there wasn't actually one).

There was no minority report for Ann Lively's murder. Instead, the vision of Ann Lively's second murder (by Burgess) was classified as an echo, which is an entirely different concept, as explained in the film:

That's just an echo. Pre-Cog deja vu, if you will. The really bad ones the Pre-Cogs see over and over again.

This is not the same as a minority report. It is basically the repetition of an old vision, just that this time it only looked like an old vision and was thus ignored by the supervising PreCrime technicians, presumably too early to realize that it actually was a different murder and showed Llamar Burgess as the perpetrator:

You made the real murder look like an echo, knowing the tech would do what he was trained to do: disregard it.

I already have ticked Ian's answer for it answers satisfactorily the second part of my question. For the first part, here are my own thoughts after some deliberation.

There are a few categories of events to distinguish among one another.

1. A unanimity event: 3 precogs see the same murder and agree on all details. Examples: (i) Howard Marks' murders of his wife and her lover (ii) Burgess's murder of Anne Lively and (iii) Anderton's murder of Leo Crow. Of course, the tech mistakes (ii) for an echo and deletes it. But, interestingly, for Agatha, it is an echo. Please see below.
2. A minority report event: 3 precogs see the same murder but Agatha disagrees on certain details. Example: none is shown.
3. An asynchronization event: 3 precogs see different murders on the same victim. Example: (iv) two male precogs see John Doe's murder of Anne Lively but Agatha sees Burgess's murder of Anna Lively.
4. An echo event: a pre-cog re-sees an already seen vision. It can happen at the group level when, for example, all 3 precogs re-see Howard Marks' murders. It can also happen at the individual level when Agatha re-sees Burgess's murder of Anna Lively.

Burgess's plan is quite ingenious. He expects first (A) a unanimity event in which all 3 precogs see John Doe's murder, then (B) another unanimity event in which all 3 precogs see his own murder, and (C) the tech classifies (B) as an echo of (A). So (B) and (C) do play out as planned but for (A), instead of a unanimity event, he gets an asynchronization event!

So to answer the question: the movie, through Anderton and Hineman, treats Agatha's missing vision as a minority report because, as far as they know when their conversation in the garden takes place, a minority report event (or foul play) is the only way that only Agatha's stream is deleted but not the streams of the male precogs. They certainly do not think of an asynchronization event, which must be exceedingly rare because the circumstances that can possibly trigger it and can confuse the precogs must be engineered in a very clever way. And that is precisely what Burgess has done.