Generally speaking, beams are a step backwards. In many cases you will have more false alarms, and it's also much harder to define the precise area of coverage.
However, you're boiling a fairly complex topic down to 2 basic components and I think the solutions are more varied than just what you have presented.
There is a rough scale that looks something like:
Outdoor PIR - Motion Detection - Beam - Advanced Motion Detection - Fence Sensors - Laser - Low-Level Analytics - Object Classification Analytics.
As you go up that scale from left to right, the products become more intelligent and generally more tunable and adaptable to challenging environments. Note that price does not scale across the board, some good analytics options are cheaper than potentially lower-grade outdoor beams. I grouped Fence Sensors into one category, though that really spans a few options as well, they're just not as commonly used these days (IME).
Roughly speaking as you up the scale you can cover more area with a given product class. If you're just looking to cover a small choke point area, an outdoor PIR might be perfectly adequate, but if you're trying to cover a parking lot that is next to a nature preserve you likely won't be successful with anything less than object classification analytics.
The two main factors that go into chossing what to use are "What am I trying to detect" and "What am I trying to ignore". The more precise you need/want to be on either of those two questions, the more advanced your solution will need to be.
For true video analytics products the differentiating factor on the technology side often comes down to how good the system is at ignoring motion from similar-sized objects, and how difficult it is to get the system tuned or setup properly.
I think that "video analytics" is a term whose definition is constantly being redefined, much like "high resolution video" used to mean anything with more than 500TVL, and now it means anything 1080p or above. For the current market, any product that can't easily be configured to not generate lots of false alarms on things like leaves/foliage, shadows, etc. is probably more in the class of motion detection than analytics. I would also define "long range" as 250'+, FWIW.