Is it correct to assume a 1/2 wire dipole for 20m will act as a regular dipole on 10?
No, at least not if I understand your question correctly.
A half-wave dipole for 20m would be a full-wave dipole for 10m. While on 20m it has
a low SWR using coax feed, a full-wave center-fed wire has a high impedance, so the
SWR may be 20 : 1 or worse on 10m.
There are three simple solutions for multi-band use:
First, use separate dipole wires for each band with a common feedpoint and coax. This
is what I do with my portable dipole kits - I carry a length of coax attached to a center
insulator with wing nuts, along with dipole wires (2 x 1/4 wave) for each band I am
likely to want to operate. Each time I set it up I attach some combination of wires
to the insulator depending on band conditions, time of day, space available, etc. Sometimes
it covers 80m through 10m, sometimes just one band such as 20m or 10m. I tie the ends
of the wires on each side to different points to separate them somewhat, and it allows
me to work multiple bands without a tuner.
The second method is to put up an antenna (such as a 20m dipole) and feed it with twinlead
or other low-loss balanced line to a tuner. With that you can operate on any frequency for
which the antenna is long enough (minimum length for good efficiency is around 1/3 wavelength.)
This is much simpler to install, being only a single set of wires, and allows many more bands
(especially for 20 / 17 / 15 / 12 / 10m) but requires a wide-range tuner.
A third method is to use some sort of multi-band antenna where the antenna is designed to
have a low SWR on some number of bands. Examples would be a trap dipole or vertical, OCFD,
G5RV, etc. This takes more work and adjustment to build (or more expense to buy) and some
types only work well over a limited number of bands, but they work well for some people.
Certainly if you were only interested in 10m and 20m, a trap dipole might be a good choice,
as it would allow operation of both bands with coax feed without a tuner and only a single
wire to string up.