Yes, a Moxon is basically a 2-element yagi with the ends bent towards each other.
I've used indoor loops and similar antennas several times. My favorite approach is to
attach it to a glass window using suction cups - they sell ones with hooks attached
for hanging Christmas lights. (My current UHF antenna is a 2-wire ground plane hanging
on the window over my desk.) If you tape the wire to the window glass it will shift
the tuning due to the different dielectric constants, but hanging half an inch away
doesn't seem to make much difference.
If you make the loop rectangular and about twice as wide as it is tall, fed in the center
of a short (vertical) side, it will have an impedance close to 50 ohms and you can feed
it directly instead of using the quarter wavelength of 75 ohm coax. It isn't as easy to
switch between vertical and horizontal polarization, however, unless you build it on a
frame of some sort. Otherwise you can use a square loop or triangular loop with the
matching section.
For true vertical polarization the delta loop is fed a bit up from the bottom corner, just
1/4 wavelength down from the top corner. This probably doesn't make a lot of difference
in your application. If you use a continuous length of wire for the loop that passes through
holes in the insulators, you can slide the wire around and shift it between horizontal and
vertical polarization without having to untie anything. In fact, if you installed the loop
with the top horizontal, you could slide the feedpoint up and down one sloping side
to get both horizontal (at the bottom) and vertical (1/4 wave up from the bottom).
Hmmm... with your current loop, if you set it up for horizontal polarization (fed at the
bottom point, top horizontal) then you could leave one of the top corners connected
and just swap the other two for vertical polarization - that would still have the top
wire horizontal and the point down, but fed in an upper corner, which gives the same
polarization as when it is inverted. That might make it faster to change polarization.
A 2-element quad pointing out the window is another step up - I build them using PVC
pipe and #14 wire because that is what I have lots of around the farm, but there are
lots of other methods that can be improvised.