Something like persistent X forwarding

Simple answer is "Yes." The ICA protocol was developed when dial-up was a very common internet connection : if you tuned it properly, you could fit a single ICA connection into about 8-10 kbps. You can add, remove, and rate-limit all the channels for things like file transfers, sound mapping, removable devices, etc.

RDP is a lot like ICA but not quite as configurable.

VNC is a remote framebuffer - in its original iteration, it was a think-tank project for an ATM LAN, when 155 Mbps was insanely good. Nowadays, many flavors of VNC have lots of different compression options, so it's better than it was.

The X session startup commands run by VNC are generally found in ~/.vnc/xstartup.

Your top screenshot looks like the old default X session of xterm & twm.

To start the GNOME desktop instead, you can insert a call to run it if it’s available, such as:

#!/bin/sh

vncconfig -iconic &

if [ -x /usr/dt/config/Xsession.jds ]; then
  exec /usr/dt/config/Xsession.jds
fi
[ -r \$HOME/.Xresources ] && xrdb \$HOME/.Xresources
xsetroot -solid grey
xterm -geometry 80x24+10+10 -ls -title \"\$VNCDESKTOP Desktop\" &
twm &

(I've not tried this on Solaris 10 myself, but have using the much newer versions of VNC & GNOME in Solaris 11, though that uses different commands to start GNOME 2.30.)