What is an extent rectangle?

I think you'll find there is a bit of overlap with these definitions. They're all very similar, in my opinion. However, ESRI has a glossary of GIS terms, so I just looked them up. The definitions are similar or identical to the wiki GIS glossary as well.

Bounding Box (Bounding Rectangle):

[map display] The rectangle, aligned with the coordinate axes and placed on a map display, that encompasses a geographic feature or group of features or an area of interest. It is defined by minimum and maximum coordinates in the x and y directions and is used to represent, in a general way, the location of a geographic area.

Extent:

The minimum bounding rectangle (xmin, ymin and xmax, ymax) defined by coordinate pairs of a data source. All coordinates for the data source fall within this boundary.

It should be noted that in the arcpy documentation for extent (though you haven't said which software you're using), the definition is essentially the same:

An extent is a rectangle specified by providing the coordinate of the lower left corner and the coordinate of the upper right corner in map units.

Extent Rectangle (you didn't ask for this one, but it was in the glossary so I threw it in):

[ESRI software] A rectangle that is displayed in one data frame, showing the size and position of another data frame.

The glossary doesn't have a definition for bounds, but I think it would likely be similar or identical to extent. It should be noted there is also a term called "metes and bounds", which is not really what you're asking, but I thought I would include it anyway.

Metes and Bounds:

A surveying method in which the limits of a parcel are identified as relative distances and bearings from landmarks. Metes and bounds surveying often resulted in irregularly shaped areas.

Finally, there is a tool in ArcGIS called Minimum Bounding Geometry which:

Creates a feature class containing polygons which represent a specified minimum bounding geometry enclosing each input feature or each group of input features.

Depending on which option you choose, the results will be different. However, it's worth noting that the Envelope option is a polygon which appears to be to be identical to the extent.

Extent indicators are a way to show the extent of one map frame within another map frame. They are often used for layouts containing locator or overview maps. A locator map shows a larger area, or extent, than the main map to provide spatial context. An extent indicator can be added to the locator map to show the area represented in the main map. Inset maps show areas of your main map in more detail. With inset maps, one or more extent indicators can be added to your main map to show the areas represented in the inset maps.

Extent indicators require two map frames—the map frame containing the extent indicator and the map frame with the extent you want to show—and only work with 2D maps. Because of this, the Extent Indicator button is unavailable until there are at least two 2D map frames on the page. To add an extent indicator in your layout, complete the following steps:

In the Contents pane, select the map frame to which you want to add an extent indicator; this is the map frame where the extent indicator will draw.
On the Insert tab of the ribbon, in the Map Frames group, click the Extent Indicator button. A drop-down menu appears listing all the other map frames on the page. Choose the map frame for which you want to show the extent.
An extent indicator draws on the map frame.
Optionally, select the extent indicator in the Contents pane, right-click and choose Properties to open the Element pane. There you can change the symbol, add a leader line from the extent indicator to the map frame of which it shows the extent, and collapse the indicator to a point.

Extent indicators are listed beneath the map frame with which they are associated. If you don't see the extent indicator in the Contents pane, expand the map frame. If the map frame contains multiple extent indicators, you can select all or some and format them simultaneously. You can also reorder the extent indicators within the map frame by dragging them into the Contents pane.

To change the extent indicator's symbol, right-click the extent indicator in the Contents pane and choose Properties to open the Element pane. Select the Symbol button and choose a new symbol from the gallery, or select Properties to modify an existing symbol.

By default, extent indicators are the same shape as the map frame of which they are showing the extent. You change the shape of the extent indicator by setting the Shape drop-down control. The following options are available:

Frame—Draws the extent indicator the same shape as the map frame of which it shows the extent. This is the default.
Rectangle—Draws the extent indicator as a rectangle, regardless of map frame shape. This may result in the extent indicator showing slightly more than the actual extent.
Current map series shape—Draws the extent indicator the same shape as the map series index feature. This is only available when spatial map series is enabled.

Check Apply symbol outside extent to have the extent indicator draw on the area surrounding the extent, rather than the extent itself. This can be used to mask features outside the extent, further highlighting the map extent being shown.

A leader can be added to connect the extent indicator to the source map frame. It can be a line or a callout. The following leader options are available:

<None>—Does not draw a leader line. This is the default.
Shortest line—Draws a line showing the shortest distance between the extent indicator and the source map frame.
Line to midpoint—Draws a line between the extent indicator and the midpoint of the closest side of the source map frame.
Line to center—Draws a line between the center point of the extent indicator and the source map frame and clips the line to the edges of the extent indicator and source map frame.
Callout to edge—Draws the source map frame as a callout by drawing a polygon between the extent indicator and the shape of the source map frame.
Callout to center—Draws the source map frame as a callout by drawing a polygon between the centers of the extent indicator and the source map frame.
Callout to edges—Draws the source map frame as a callout by drawing a polygon between the corners of the extent indicator and the shape of the source map frame.

To add a leader, right-click the extent indicator in the Contents pane and select Properties to open the Element pane. In the pane, under the Leader heading, set the Style to the desired option. To change the leader's symbol, click the Symbol button and choose a new symbol from the gallery, or select Properties to modify the existing symbol.

Sometimes the area displayed in the source map frame is so small on your main map frame that the extent indicator is better symbolized as a point. The collapse to point option allows you to set the smallest width or height value in points where the extent indicator should be drawn as a shape. If the width or height of the extent indicator is smaller than that value, the extent indicator draws as a point.

To set collapse to point, right-click the extent indicator in the Contents pane and select Properties to open the Element pane. In the pane, under the Collapse to point heading, set the Smaller than control to the smallest width or height value where the extent indicator should draw as a shape. To change the symbol used when the extent indicator is collapsed to point, click the Symbol button and choose a new symbol from the gallery, or select Properties to modify the existing symbol.

When the extent indicator draws as a point and Apply symbol outside extent is checked, that setting is ignored and the point draws on the map extent, not outside of it.