The videos both appear to be using amateur rocket engines, the type that can be bought over the counter in a modelling shop, or one that specialises in aircraft models and amateur rocketry.
The fact that these, which are built with a solid propellant, are difficult to start up in a vacuum, is not of itself a proof that all rockets, of all types, behave in a similar way. The ones used here appear to rely on an electrical starter (certainly those I've used in the past have done this). Once started, even they can clearly be seen running in the vacuum.
The fuel used in, for instance, the Lunar Module, was hypergolic (DiNitrogen Tetroxide and AeroZine 50). It needs no external starter, and relies simply on the combination of two fuel elements. Place them in contact with each other, and combustion results, with no ignition process or system. This is a different kind of engine to the amateur ones above.
https://en.wikipedia.org/wiki/Hypergolic_propellantAs a for-instance of another type, the SpaceX Falcon 9 second stage uses a Merlin engine fuelled by a combinaton of RP-1 and liquid oxygen, both of which are a bit out of the reach of the amateur.
The ignition system uses a TEA-TEB as an igniter fluid. The mixture is pyrophoric, and ignites spontaneously when it comes to contact with oxygen (including the liquid supercooled oxygen used in the primary fuel mix). Essentially, throw some of this into the path of the primary fuel mix, and this lights it up.