Similarly to many commenters here, I actually don't like this word much. Personally, I would probably use video capture. Individual tastes aside, however, screencast is a term that is very well attested in dictionaries, books, and software, and if you look at the evidence below, seems to be preferred overall.
Let's start with the
Collins Dictionary (with thanks to Edwin Ashworth):
a digital recording of the display on a computer screen that may be viewed on another device
a digital recording of computer screen output
a digital recording of a computer screen's display
A video recording or transmission of the data displayed on the screen of a computer or mobile device, typically with accompanying audio.
‘this tutorial contains a screencast demonstrating all of the steps’
verb (no object)
Record or transmit video of data displayed on the screen of a computer or mobile device, typically with accompanying audio.
On the other hand, screencast is not recorded in the OED, Merriam-Webster, American Heritage, or Cambridge.
Here is the lede of the Wikipedia article "Screencast":
A screencast is a digital recording of computer screen output, also known as a video screen capture, often containing audio narration. The term screencast compares with the related term screenshot; whereas screenshot generates a single picture of a computer screen, a screencast is essentially a movie of the changes over time that a user sees on a computer screen, enhanced with audio narration.
True, in general, one should not take Wikipedia as a reputable source. However, my experience is that tech-related topics are usually treated quite authoritatively there, at least once the article has existed for a number of years (as this one has).
Use in published literature
Here are some books that use that terminology:
Screencasting for Libraries (source)
The Use of Screencasting in Higher Education: A Case Study (source)
iPad in Education For Dummies
Screencasting on desktops and laptops has been around for a number of years. Screencast software records whatever is displayed on your screen and enables you to add a voiceover narration during the recording. (source)
Sixth Grade Technology Curriculum: 32-lesson Comprehensive Curriculum
Have screencast and screenshot tools available as software, web-based tools, or add-ons. (source)
Use in software applications
Here are some examples of video capturing applications that use that term.
Kazam on Linux:
An example from Adobe:
Some other examples of software that uses this term: Jing, Screencast-o-matic, Bandicam…
Facebook uses the term screencast a lot in its pages for developers, i.e. here
Next, you upload a screencast of your app and how it is using the permission.
You can use any of these accounts to test your app and create a screencast.
How can I test my app and record a screencast?
Provide a screencast.
Provide a Detailed Screencast and Clear Instructions = Quicker Evaluation
Our review team examines your screencast and follows your usage instructions to reproduce the personalization elements of each permission. Providing a detailed screencast and clear instructions will result in quicker evaluation of your submission.
(With thanks to Andrew Leach.)