The density matrix is determined for H(n=3) atoms produced in axially symmetric electron-transfer collisions of 80-keV protons on helium. In the experiment axial or transverse electric fields with respect to the proton beam are applied to the collision region. The intensity and polarization of Balmer-α radiation emitted by the H(n=3) atoms are measured as a function of the strength of the external electric field. Detailed analysis of the measured optical signals, taking into account the time evolution of the H(n=3) atoms in the applied electric field, makes it possible to extract the complete density matrix of the H(n=3) atoms at the moment of their formation, averaged over all impact parameters. Significant improvements in the experimental technique and in the data analysis associated with the fit of the density matrix to the optical signals have eliminated systematic effects that were present in our previous work [Phys. Rev. A 33, 276 (1986)]. The improvements in the apparatus are as follows: application of electric fields using electrodes with a simple geometry for the axial and transverse orientations that allows accurate calculation of the spatial variation of the electric field inside the collision chamber; use of high-quality optical elements and a rotatable, single-unit design for the polarimeter; automated gas handling for background subtraction; and full computer control of the electric fields, polarimeter, gas handling, and data acquisition. The analysis incorporates the following improvements: hyperfine structure of the H(n=3) manifold; cascade from the H(n=4) manifold; nonuniform detection efficiency over the viewing region; and modeling of the nonuniform electric fields, the nonuniform gas density, and the exponential decrease of the proton beam current in the gas cell due to electron transfer. With these improvements the results from axial electric field measurements are in good agreement with results obtained independently from transverse electric fields. Moreover, the extracted density-matrix elements are found to be within their physically meaningful bounds. The major results from 80-keV collisions are that the H(n=3) density matrix has an average coherence of 81%±1%, an electric dipole moment of 3.50±0.09 a.u., and a first-order moment of the electron current density distribution 〈(L×A)z,s〉 of -0.13±0.02 a.u. Results from a recent calculation show qualitative agreement with the experiment.