The crystal must pass the feedback to the base of the transistor with right phase shift at the right frequency. If this oscillates, it happens. Obviously the oscillation frequency is near the serial mode resonant frequency of the crystal.
The tapping and transformer structure of the resonant circuit is developed to keep the oscillation waveform sinusoidal enough (=low enough transmitting power at the harmonic frequencies). Your own oscillator examples do not pay any attention to harmonic content.
Tuning the 365pF capacitor needs something. One can add a miniature incandescent bulb (=a lamp with filament for a small flashlight) in series with the antenna wire.
ADD due the comment:
Sinewave oscillators are amplifiers with feedback. They oscillate just like a sound system will scream when one brings the microphone too near the loudspeaker. The oscillation happens at the frequency where the signal returns from the output back to the input through the feedback path just in the same phase angle (=delayed full cycles) as it was inputted to the amp and at least as strong or amplified.
The crystal causes wildly varying phase shifts in different frequencies. Proper crystal oscillator designs have the crystal in the place where the oscillation conditions come true only in one frequency. In the mighty transmitter the feedback from amp output to the amp input is strong enough and causes the right phase shift near the series resonance mode frequency of the crystal.
Harmonics: A pure sinewave is a dream. Practical oscillators distort the waveform. The well known math of Fourier series or a practical test with a transmitter and radio receiver proves that practical oscillators send at a series of frequencies. If the crystal is for 7MHz, there's also some output at 14, 21, 35, 42, 56...Mhz, all at the same time. The resonant circuit at the output can be designed to be as well a part of the feedback circuit and a bandpass filter which attenuates the unwanted frequencies weak enough to stop collecting complaints.
Tapping and transformer make possible the resonant circuit to have at the same time low enough attenuation for the transmitting frequency and high enough attenuation for other frequencies. Proper design needs complex variable mathematics and it was seriously researched already when electron tubes were taken into the radios.
Tuning: This circuit works only if the output filter is tuned to not attenuate dead the frequency of the crystal. You need some indicator of existing output. Adjust the capacitor for maximal current in the antenna wire (=brightest light in the lamp in series). The right adjustment is critical.
Your own example oscillators: The leftmost has the feedback through the crystal. The rightmost is a little tricky. The transistor is used as common base amplifier, the input is at emitter. Feedback is through C1. The crystal in this circuit acts as an parallel resonant output filter and the harmonic content is obviously lower than in the leftmost circuit.
An link to general oscillator theory (no crystal oscillators, all applications are for low frequencies with TI's parts)