To make sure there is resonance, you need to look a bit around the points of real impedance.
First, imagine a load made of lots of passives, Rs, Ls and Cs. Combining them in series and parallel, you wind up with an equivalent load ZL whose value is a formula dependent on the frequency, some rational function p(s)/q(s), where s is the complex frequency s = I·w.
Imagine that, for some (angular) frequency w0, that load impedance is a real number. Why can you consider that there is a resonance? Start with the fact that the load impedance at that frequency is \$Z_L(\omega_0) = R_L\$, some real number. That isn't saying too much: we only know that at that very precise frequency you have something like a resistor load. So let us see what happens at nearby frequencies. Since rational functions are smooth enough, near the resonat frequency the impedance must be, to first order, a small variation of that real load:
\$Z_L(\omega_0 + \mathrm{d}\omega) \ = \ R_L + (a + bi)\mathrm{d}\omega \$
where a and b are some real numbers. Further, imagine that b is not zero (that would be a rather singular case). If we aggregate things a bit, near the resonant frequency we have:
\$Z_L(\omega_0 + \mathrm{d}\omega) \ = \left(R_L + a\mathrm{d}\omega\right) \, + \, ib\mathrm{d}\omega \$
What this formula is saying us is that, near the resonant frequency, as frequency rises impedance goes from capacitive to inductive (if b>0), or as inductive to capacitve (if b<0). If b is a big number, that transition will be very fast: this is somehow related to the Q of the resonance.
So a resonant frequency can be considered a point where load impedance goes from capacitive to inductive. How is this related to oscillation? Imagine that load is connected to some non-inverting gain element, for example, the virtual ground of an operational amplifier. If the gain is big enough, at resonance the opamp will feed back a signal in phase and cause oscillation at frequency w0. However, as you move away from w0, the phase shift caused by the increasing reactance (the i·b·dw term) will make oscillation impossible. You get a sinusoidal oscillation only at frequency w0.
The low pass filter of the previous posts is not a great resonator because the transitions were really slow: impedance was real (about 100 + 0j) at 162MHz, and reached about 290 + 79j ohms at 200MHz. That's a slope b = 79/(200-162) = 2 ohms/MHz! The 200MHz resonance is a bit better, but compare that to a crystal resonator, who can have slopes of hundreds of ohms per Hertz. Now that is resonance.
To summarize, just considering an LC network for its resonances is a restricted point of view. You get a convoluted variation of impedance with increasing frequency, and you can achieve a lot more than selecting a very specific frequency with that behavior. Besides, as MrAl said, there are other points of view about resonance, for example, tracking the stored energy in the system when you inject a sinusoid into it. A good part of the problem is defining resonance of the LC network in uninequivocal terms.
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