  We study stars in the J-regions of the asymptotic giant branch (JAGB) of
near-infrared color magnitude diagrams in the maser host NGC 4258 and 4 hosts
of 6 Type Ia supernovae (SN Ia): NGC 1448, NGC 1559, NGC 5584, and NGC 5643.
These clumps of stars are readily apparent near $1.0<F150W-F277W<1.5$ and
$m_{F150W}$=22-25 mag with James Webb Space Telescope NIRCam photometry.
Various methods have been proposed to assign an apparent reference magnitude
for this recently proposed standard candle, including the mode, median,
sigma-clipped mean or a modeled luminosity function parameter. We test the
consistency of these by measuring intra-host variations, finding differences of
up to $\sim$0.2 mag that significantly exceed statistical uncertainties.
Brightness differences appear intrinsic, and are further amplified by the
non-uniform shape of the JAGB luminosity function, also apparent in the LMC and
SMC. We follow a 'many methods' approach to consistently measure JAGB
magnitudes and distances to the SN Ia host sample calibrated by NGC 4258. We
find broad agreement with distances measured from Cepheids, tip of the red
giant branch (TRGB), and Miras. However, the SN host mean distance estimated
via the JAGB method necessary to estimate $H_0$ differs by $\sim$0.19 mag
amongst the above definitions, a result of different levels of luminosity
function asymmetry. The methods yield a full range of $71-78$ km s$^{-1}$
Mpc$^{-1}$, i.e., a fiducial result of $H_0=74.7\pm2.1$ (stat) $\pm$2.3 (sys)
($\pm$3.1 if combined in quadrature) km s$^{-1}$ Mpc$^{-1}$, with systematic
errors limited by the differences in methods. Future work may seek to further
standardize and refine this promising tool, making it more competitive with
established distance indicators.
