The first impulse response looks very much like a minimum-phase response. I don't think that you'll get away with half the number of filter taps for the minimum-phase filter. You might have a slightly shorter filter compared to the linear-phase filter, but the difference is usually small.
Note that there are infinitely many filters with exactly the same magnitude response. Linear-phase and minimum-phase filters are just two specific examples. Obviously, the difference between all these filters is the phase response. All filters except the linear phase filter introduce phase distortions. The minimum-phase filter introduces the smallest overall delay.
Practical linear-phase filters are always finite impulse response (FIR) filters. Minimum-phase filters can be either FIR or IIR (infinite impulse response).