Purpose of the different filter panels

This is a good question that requires a long answer!  From my experience deploying Pyramid, it is an area that some people initially misunderstand. At first, we thought it was overly complicated. But now we understand it and cannot imagine not having all these options. In a nutshell, Pyramid has numerous mechanisms for filtering data – and once you understand how they all fit together, you’ll find a deeper appreciation for the platform’s sophistication.

To start, we need to split filters into 2 types: numeric filters (top count, above x) and data filters (filter by country, filter by product).

Numeric Filtering

These numerically filter results via various entry points: simple quick filters in the right click menus; drop zones (when using measures); and the more advanced filter wizard in Discover (#2 in your list). They are not strictly limited to numerics only, as they can include text based filtering and logic that can include data elements in driving the numerical values.

Data Filtering

These use data to limit what you see in your results. They are often called “slicers” and are created in Discover through the  Filter dropzone, where you add attributes and hierarchies to the visualization or dashboard to produce lists (#1 in your list). Users can then pick different elements from those lists to filter the report in Discover, Present and Publish.

Data can also be filtered in Discover via the element trees (#3 in your list) – which is a listing of all the items in an attribute (countries in the Country attribute etc). As per your question, the element trees in Discover are similar to the slicers, in that a user would pick what they want to see from the element lists (or trees) and it would then filter the report accordingly. In that way, the element trees are part of the data filtering side. But, they tend to be more about static selections, useful to the users of Discover only. If you wanted to let an end user access these element lists, you’d show them as slicers (as per above).

Regardless, both element trees and slicers have their own localized search capabilities, allowing users to find (and filter!) what they are looking for within the various lists.

BTW: One big mistake we made early on, was to expose every attribute as a slicer option in Present dashboards, because non-technical users don't have access to the element trees in Present. We now understand that this is confusing to those users. We're far better off showing users focused analysis and a smaller set of options. Less is more. 

Background vs Slicer

The background filter (#3 in your list) is just a single element data selection to filter your query. It only shows up, if the attribute is not directly visible in a drop zone – so it’s a background filter. If you chose multiple elements, it would generally become a slicer. It’s a simplifying, space saving UI concept.

Datapoint Selections

When you use the focus on data point context menu (#4 in your list), it does a convenient multi-way element selection from each of the relevant attributes, and limits the visual to just those selections. Think of it as a fast way to make multiple element tree selections. The eliminate function does the opposite, by removing the elements from the list.  (BTW: You can use CTRL+click to do multiple datapoints in a single move). This can get even more interesting, if you attempt this same idea off the elements in a visual themselves, rather than the data points (or data cells). Here you can eliminate elements and also choose to eliminate combinations of elements – creating ragged queries structures.

Cross-over points

With all that said, here are some other things to consider:

1. It’s possible to create a numerical filter by dropping a measure into the filter dropzone. This is just a convenient way to create a numeric filter. Most of the time, the filter drop zone is used for building slicers in Discover. In Present and Publish, you can re-use the slicers or build them from scratch.
2. When users right click on an element, and "Dice" it, they are effectively drilling down into the new attribute/hierarchy and filtering the new query by the triggering element. So in a query of country sales by year:  if the user dices the "USA" in the country attribute by product, the resulting query is a list of product sales by year, filtered to the USA. 
3. The List or Set builder in Formulate lets designers build incredibly complex lists that can be used to drive slicers or selections. The list build blends data filtering and numeric filtering to produce smart, dynamic lists that in turn can be used like everything else. The independent lists, created in Formulate, can be used as selections to drive slicers or visible element selections in the visual in Discover (instead of hand picked elements from the element tree); drive slicers in Present; or drive Publication reports.
4.  Parameters, another advanced Formulate tool, lets you create lists that drive other lists or formulas or selections. They too are a filtering mechanism in a way.

So, yes there is a lot. They all do different things. Hope it clears it up a bit.