First thing I would suggest is that you look at stereo recording methods, which I think you really need to utilise to record a soundscape.
Basically there are various options - AB, XY, ORTF and Mid-side are probably the commonest. Most methods use different types of mic, so probably best to choose a method first, rather then buy mics and find that you are limited to a method you don't like.
AB will probably create the best stereo image, as the two omni mics are separated (rather like your ears), so there is a time difference between the signal reaching each mic. The wider you space the mics the greater the stereo effect, but in any case the mics need a fair separation, so it is not the most portable option. Also if you do need to create a mono version of the recording, you cannot easily mix the two tracks (as the separation may create phase issues).
With XY, two cardiodal mics are angled towards each other, so the stereo image is created by a difference in loudness in each mic (but not a difference in the time the signal is received). This is a bit of a fudge and generally creates a less immersive stereo recording. This is the method by which your H5 creates a stereo recording. A near-coincidental stereo recording can be mixed into a mono recording. As the mics are cardiodal, you will pick up less ambient noise from behind the mics - so good if you wanted to record say a dawn chorus from the woods in front, but don't want sound from behind you. [Is there a reason you don't like the results from the H5? - I suppose a downer with using in-built mics (and/or the XY method) is that you don't really have much control over the wideness of the stereo effect, whereas with other methods you can change the apparent 'width' of the recording.]
ORTF is basically a combo of the above two. So two cardiodal mics are separated and angled two each other (but this time facing away from each other) and also separated. This means there is a difference in loudness and also a separation in the signals reaching each mic. The combined effect, means that you can generally have less separation than an AB setup, but still get a good stereo recording. I would use this option, if your area happy to invest in two mics and willing to carry all the necessary equipment into the field. Some recorders purport to use this technique, but the separation of the internal mics is then less than advised, so the stereo effect is then probably not much different to a recorder with XY mics.
Mid-side is a bit of a specialty, and you need a recorder that can decode an MS signal. With this you have a cardiodal or super-cardiodal mic (I think you can even use an omni?) combined with a figure-of-8 mic. By changing the gain on the figure-of-8, you change the perceived wideness of the stereo. Again it is near co-incidental, so difference in loudness is creating the stereo effect. In theory mid-side can be a very compact set-up, but in my limited investigations it seems an expensive option - particularly if you need to change wind-shields, suspension systems etc. to house two parallel mics.
Obvious thunderstorms and soundscapes are 3D, whereas stereo is really a 2D effect, presumably designed to recreate the 2D arrangement of music (with the stage in front). Binaural recordings can recreate a 3D effect, but only when listened to through headphones. To create a binaural recording you would either need a binaural set up (basically a dummy head with two mics positioned as per your ears), or you could record with an ambisonic mic and then render to binaural with appropriate software. Ambisonic mics generally need a 4 track recorder with ambisonic decoder. The Mix-Pre 6 is an option, but with the ambisonic plug-in you loose 32bit float capability - which would be a downer for recording thunderstorms, when it may be hard to pre-judge recording levels. With the stereo options above you could retain 32bit float capability.
I wouldn't use a shotgun or parabola for soundscapes as this is not what they are designed for. Even a stereo parabola is really designed to capture a focal sound with a stereo background - akin to a mono recording with high gain mixed with a background stereo recording with no or little gain.
Once you have decided on a method, perhaps someone can advise on appropriate mics to suit the chosen method (though I think most of us are here are recording birds [in mono] with directional mics).