Technical Term for "side by side scene editing"?
What you are describing is called a "ripple edit" (because the adjustment you make to the clip ripples through the rest of the timeline that comes after it).  It may also be called something like "ripple trim" as a more specific term for the precise type of ripple edit.  It is a feature available in any decent professional editing package.  Depending on your other needs, it's probably worth taking a look at DaVinci Resolve as Resolve is a high end professional product which also offers a completely free version with about 90-95% of its features included.  It's out of your price range for the full version (at $299 for a license that they haven't charged for an upgrade in the last 3-4 versions) but there is a decent possibility it will cover your needs in the free version.

There is lots of other software with Ripple Edit capabilities, but Resolve is the cheapest one I personally work with enough to be able to recommend.  I've been out of the consumer market for a very long time, so I'm not sure what products would be current there.