This picture (from the document linked below) shows the Vanguard engines; the first stage engine is in the rear.
The engine was an X-405 built by General Electric. The launch vehicle was built by Martin Marietta. These two contractors differed on the exact cause of the explosion, as the question states.
The Martin people traced what they called an "improper engine start" directly to a low fuel tank pressure which was responsible for a low fuel injector pressure prior to the start of the turbopump operation.
The low injector pressure allowed some of the burning contents of the
thrust chamber to enter the fuel system through the injector head.
According to this version of the accident, fire started in the fuel
injector before liftoff, resulting in destruction of the injector and
complete loss of thrust immediately after liftoff.
They traced the immediate cause of the explosion to a loose
connection in a fuel line above the engine. Their reading of the
telemetered and photographic data was that there was no "improper
start." On the contrary, the engine had come to full thrust, only to
lose thrust when a little leaked fuel on top of a helium vent valve
blew down on the engine.
Quotes from Vanguard - A History p. 210
I think what they are talking about is a "burp" where the chamber pressure in the engine gets higher than the feed system pressure and hot gas flows upstream into the propellant system. In this case the reversal of the desired pressure gradient seems to have been caused by the feed system. Maybe the turbopump cavitated?
Regardless of the cause, the engine (or feed system) explosion was relatively minor but it resulted in a complete loss of thrust almost immediately after liftoff. The vehicle fell almost vertically back down and crumpled; all the propellants spilling down and being ignited by the already-burning wreck of the engine caused the "spectacular explosion".
You can see the engine (or feed system) explode at 0:28 in the second video, then the vehicle falls, then the big bang at 0:31.
This schematic of the first-stage propulsion system is from the same text.