I will try to summarize these specific circuit solutions into a "philosophy" of the simplest possible relaxation oscillator.
In such an arrangement, some substance like water, air, sand, data, money, etc. accumulates in a tank and its level is constantly rising (it is moving in one direction). In our case, this is electric charge (potential energy) in a capacitor. It is charged by a voltage source (through a conductive path in series) so its voltage "moves" towards the positive supply rail. Finally, it approaches the rail and stops there. The problem is, "What do we do to keep this movement going forever?"
The solution is to reverse the movement direction (just like we swim back and forth in a swimming pool) by discharging the capacitor. We can do it by connecting another conductive path in parallel to the capacitor (even without disconnecting the charging path). The voltage will begin "moving" towards the negative supply rail. When it approaches the rail, we reverse its "movement" by charging the capacitor again...and so on so forth...
To automate this arrangement, we need a switch with memory that is toggled when the voltage reaches the supply rails (in a manner the end switches control a motorized curtain). It can be implemented in a few ways. Some of them are more sophisticated as in timer 555 built by threshold comparators, latch and transistor switch. Other are simple, e.g. consisting of a 7414 Schmitt trigger and RC circuit. But the "elegant simplicity" is to connect a 2-terminal element combining memory and switching functions in one. What is this mysterious "2 in 1" element?
It behaves as an "overacting dynamic resistor" which resistance, in some regions, significantly depends on the voltage across it. Initially, at low voltage, it has relatively high resistance. The capacitor charges and the voltage across it increases. Then, at some voltage level, the resistance sharply decreases in an avalanche-like manner... and stays in this state until the capacitor discharges and the voltage reaches the low voltage threshold. Then, the resistance sharply increases and the capacitor begins charging again...
This element is known as a negative differential resistor with S-shaped IV curve. When driven by voltage, it has such a behavior of a Schmitt trigger. Simply speaking, it is a dynamic resistor with memory (aka hysteresis). The neon lamp is an example of such an element with S-shaped curve.
Maybe, it would be interesting for you to understand how this magic element "jumps" when switching (it is not well explained in sources). Look at the two pictures below. To show in detail the mechanism of operation, two separate graphs are presented. The first is for the case when the voltage across the element increases; the second is when it decreases (elements with hysteresis has different behavior depending on the direction of the input change). When superimposed, the two partial curves compose the whole hysteresis curve.
In this mode, there is in total three intersection points of the two superimposed IV curves: the middle point is unstable; only the end points are stable. The IV characteristic is a multivalued function and the output quantity can take only the end stable values. The switching between the two states is an avalanche-like process accelerated by the intrinsic positive feedback. Beginning from the one end value and "looking for" the equilibrium state, the negative resistor changes vigorously but in the "wrong" direction its instant resistance. Thus it recedes further and further from the equilibrium point in an avalanche-like manner and finally reaches the other end value.
Increasing voltage (Fig. 1). Look at the IV curve (blue) of an S-shaped NDR driven by a voltage source (red). When increasing voltage reaches VH, the instant resistance decreases momentarily. Its IV curve (orange) rotates counterclockwise; the operating point A moves up ("jumps up") along the voltage source IV curve and pictures this vertical part of the curve. Thus during the jump, the current increases instantly (jumps up) but the voltage stays constant.
Fig. 1. S-shaped NDR driven by increasing voltage
Decreasing voltage (Fig. 2). When decreasing voltage reaches VL, the instant resistance increases momentarily. Its IV curve rotates clockwise; the operating point A moves down along the voltage source IV curve and pictures this vertical part of the curve. During the jump, the current decreases instantly (jumps down) but the voltage stays constant.
Fig. 2. S-shaped NDR driven by decreasing voltage