That is correct! The piano (normally) is not included in the "trio" or "quartet" count.

Flute, harp, and cello.

Karl Friedrich Abel has written:

'Quartet in A for flute, violin, viola and cello' -- subject(s): Quartets (Flute, violin, viola, violoncello)

'Six concertos for flute and strings, opus 6'

Typically a string quartet is two violins, one viola, one cello.

There are a few works written for Piano Quintet with the instrumentation of Piano, Violin, Viola, Cello and Double Bass, the most famous being Schubert's 'Trout Quintet', also known as his Piano Quintet in A major.

There are also a few works for String Quartet with Double Bass such as Antonín Dvořák's String Quintet in G major, Op.77 and Wolfgang Amadeus Mozart's Serenade in G major, K.525. Slightly smaller string works with the double bass include six string sonatas by Gioachino Rossini, for two violins, cello, and double bass.

Larger works that incorporate the double bass include Beethoven's Septet in E-flat major, Op.20.

Others include Serge Prokofiev's Quintet in G minor, Op.39 for Oboe, Clarinet, Violin, Viola, and Double Bass; Erwin Schulhoff's Concertino for Flute/Piccolo, Viola, and Double Bass; Fred Lerdahl's Waltzes for Violin, Viola, Cello, and Double Bass; Mohammed Fairouz's Litany for Double Bass and wind quartet; Mario Davidovsky's Festino for Guitar, Viola, Cello, and Double Bass; and Iannis Xenakis's Morsima-Amorsima for Piano, Violin, Cello, and Double bass.