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I want to measure the vibrations of a solid object (a tuning fork, a bone conducting headphone, the chest during singing) and to isolate them from the vibrations of the air. So I am looking for a meter that will pick those vibrations only when it's in contact when the vibrating object (hence a meter that just reads frequency probably won't work for me)

I've tried to use a smartphone's accelerometer but it didn't pick anything. I've also to use a phone's sound meter in order to see whether it's picks more sound when in contact with such object but the reading was the same.

You might want to look at contact pickups designed for acoustic musical instruments.

Try searching for contact microphones of contact transducers. These tend to be based on piezo electric devices and sense vibrations through direct contact as opposed to acoustic microphones whcih generally use some sort of diaphragm.

An accelerometer is the correct choice, but the smartphone accelerometer gets its readouts smoothed out in software before being made accessible to userspace. It won't pick up such fine vibrations. You'll need a microcontroller (for up to ~1khz frequencies) or a DSP (higher), preferably on a development board, to perform the sampling. (note you should have the sample rate at least a few times higher than the frequency you measure).

Another option is a strain gauge. This again should be read by a microcontroller/DSP, or an oscilloscope (may need some basic circuitry to provide power), although it may not perform so well on soft surfaces, like human chest.