Berean Study Bible

I do not understand.

You must have purchased it at $0 in order to get the download of an update. Check your order history.

Is this as engaging, critical and technical as other study Bibles

You have to read and evaluate it for yourself. The Bereans have been around for about a century as a group that split from the Christadelpians. They've been around since the late 18th century as a Scottish Presbyterian offshoot. Then there is the Bible Student movement groups that have used the term Bereans since early 1900's ... I suspect the Berean Study Bible is of this last group - think restorationist, Russell-influences ... Summary: The term Berean has so many meanings that the title of the volume doesn't say much about what it is, but all of the groups are serious Bible students whose work is worth having at least a passing familiarity with. As to it engaging, critical, technical nature, like any study Bible you have to read it and measure it against your own presuppositions. 

Is it the Berean Study Bible, or the Berean Standard Bible?

I think the title on the Logos product page is wrong, actually. I think it should be the "Berean Standard Bible" since the resource is only Bible text, not study notes. And since the product, when you open it in Logos, is the "Berean Standard Bible."

The Berean Standard Bible is a decent and open translation that was not "crowdsourced" but they did have an open feedback phase, if I understand correctly. More info: https://bereanbibles.com/about-berean-study-bible/ 

I'm not going to use it for my primary translation any time, but it is a useful translation that can provide an alternative. Particularly if one doesn't have one of the more modern "literal" English Bible translations (I'm thinking NASB editions and LSB here primarily.)

Is it the Berean Study Bible, or the Berean Standard Bible?

I think the title on the Logos product page is wrong, actually. I think it should be the "Berean Standard Bible" since the resource is only Bible text, not study notes. And since the product, when you open it in Logos, is the "Berean Standard Bible."

The Berean Standard Bible is a decent and open translation that was not "crowdsourced" but they did have an open feedback phase, if I understand correctly. More info: https://bereanbibles.com/about-berean-study-bible/ 

I think it used to be called "Berean Study Bible" in spite of not containing what we came to expect from a study bible, especially study notes. The about page you linked says:

"The Berean Study Bible represents a single tier of the Berean Bible. This printing contains the full Berean Bible text, footnotes, section headings, and cross references. It is not what is considered a traditional study Bible, as it includes only the text, cross-references, and footnotes."

It seems that at one point quite recently they decided to rename it as "Berean Standard Bible" (keeping the BSB abbreviation) in order to better manage expectations - and possibly to come up with study notes sometime in the future. Thus the Logos product page was correct when it was created, but should be updated to reflect the name change.