Does "a signal is buried in noise" mean that the noise amplitude is still smaller than the signal amplitude? (Special case: Lock-in amplification)

What you're missing is the bandwidth, both of signal and noise.

If you look at, let's say, a 1 V rms sinewave signal, together with 10 V rms noise on an oscilloscope, you'll see only noise.

However, if the noise occupies a 1 MHz bandwidth, and is flat with frequency, and you pass the signal + noise through a 1 kHz bandwidth filter centred on the signal, then you will eliminate 99.9% of the noise power, dropping its amplitude to 0.3 V rms. The signal will then be clearly visible.

A lock-in-amplifier is a neat way to make a very narrow filter centred on the frequency you feed in as the reference.

You can use the same principle even without sine waves. Spread spectrum systems like CDMA and GPS use a pseudo-random square wave signal as the reference, and call the 'multiply and average' process convolution or correlation. As long as the reference is is the same as the underlying signal, and as long as the averaging process produces an effective bandwidth small enough to drop the noise power, the signal can be 'dug out of the noise'. A lock-in-amplifier is a special case of the more general 'correlation with a reference' that's used for CDMA.

NASA will acquire distant, or weak, satellite signals, buried in the noise and having some frequency uncertainty, by sweeping the receiver over the range of expected frequencies.

Once acquired, such systems can tighten the Phase_Locked_Loop bandwidth even more, as long as the Transmitted signal has low phase noise.

=========================

Part of the challenge of such circuits/systems, given the need to implement an almost PURE mathematically exact CORRELATION, is the DISTORTION of the mixer or however the internal signal_model and the real signal_plus_noise are processed to generate the "We have a correlation event".

the signal amplitude still needs to be larger than the level of noise

For an LIA to be effective, the signal amplitude in its bandwidth of interest needs to be somewhat bigger than the prevailing noise in that same bandwidth.

When viewed on a scope the signal may still appear to be "buried in noise" but not if you applied a tight band limiting filter. Then the signal would be much more clearly represented on your scope image. That is something along the lines of an analogy to a LIA.