Lots of people have suggested muting the 6th string with a finger / thumb / whatever is sticking out around the requisite area at the time. This is great advice.
After 25 years of playing, I have found this to be something I rely on heavily. I'd also add that it's not always helpful to play all strings on a chord, even all the 'valid' ones. Often it's fine to leave a few strings out, making sure you hit the important ones resoundingly. If you think of it that way, playing a subset of the C chord and either missing or muting the rest starts to look easy and attractive.
Also, I have find ways to play chords such that it suits the strumming style. For example I tend to play C as an A-shape chord, shifted up three frets. I don't play the 1st or 6th string because they're damped by my fingers. This suits me very well- and if I'm borrowing a guitar, sometimes the traditional C shows up weakness in the intonation of the guitar whereas an A chord shape tends not to.
I play a G with my thumb over the 6th string, and a bit extra poking over to mute the A string. Don't need it.. so I don't play it. Purists may throw their hands up in shock. Yeah well .. it leaves the rest of my fingers available for riffs etc while playing a G.
If I'm doing fingerpicking or something that relies on the traditional C chord etc then fine I can do that - but with finger picking it's much easier to miss the strings you don'e want ( that's the whole idea! )
Regarding upstrokes generally: I notice that people very much underestimate the trickiness of the strumming hand. It's easy to concentrate on the fret hand as that's where the detail is, but the strumming is easily important as that's more to do with the way you play whatever notes you've fretted. It's important and not necessarily easy.
Are you missing something ? Really it's down to damping the string(s) you don't want played with a finger etc, and 'aiming' or biasing the strum away from the string you want to miss. Eg on the C chord you mention, to miss the 6th string, aim the strum at the top/middle 3 strings. You'll hit those more forcefully (careful though), and doubtless get a couple of extra ones, meaning maybe you'll probably play all 5 thinner strings, because it's hard to be that precise with regularity. But with muting and this biasing, you should be able to stop the unwanted strings from playing.
As an aside ..
In your pictures of the "Previous Problem" it looks like your guitar has a flat neck nylon-string ? Some guitars (mainly electric or steel strung acoustic) have a curved neck surface so that the middle of each fret is raised. This makes it a bit easier to pick out which strings you want to hit as you strum, as some poke up a bit more than others. It also can make it more difficult to get all 6 though.
On a flatter neck, I can see that you might have some trouble missing some strings when strumming. I just wanted to mention that it's not always that difficult/the same on different guitars.