A casting rod has the trigger underneath it near the reel that you hook your finger on to cast it so it doesn't fly out of your hand. Casting reels such as baitcasters (low profile and round, both) use these, as well as spincaster reels like yours is. The guides on these face upwards when in use.
Spinning rods usually have much larger guides on them, especially the first one near the reel. They do not have a trigger on them. The guides on these hang down below the rod when in use.
You may look through a bass pro shops catalog and see listed for sale what they call a casting rod AND a trigger rod. The difference there is that a casting rod has what they call a pistol grip, for just one hand. The trigger rod has a long handle behind the reel for two handed casting. That's the only significant difference there. Generally when someone says a casting rod, they can be referring to either of these, short handle or long, but it goes without saying that it will have a trigger on it.
I'd be willing to bet if you bought the Zebco Hawg 733 as a combo (rod and reel both) that the tag on the rod you bought said 20# because the reel came prespooled with that. If they were bought separately and the tag still said 20# but the rod says 8-17 then go with the rod's ratings itself. The smaller line will cast better for sure. Sounds like the 2 rods you have are both going to be medium in power, based on the one saying 8-17 line, that's a general medium line rating for most rods, and the other rod having an M in the model number, I'd bet it's a medium also. A lot of rods that are medium heavy would have an MH in it instead.
But you basically have a Zebco 33 reel (a classic for a lot of guys, they swore by these for many years), which I'd put 10# or even 8# on, and the other is a heavier Zebco Hawg 733 so you could go heavier on it if you wanted, but I'd still probably not put any heavier than 15# on it, and likely just 12# only....if it were mine. Performance should improve a lot that way, much more than the 20# line on it now.
Lure weight is a reference to the sizes of baits that a rod can cast effectively without being overloaded by it, and also still put it out there a good distance. The 1/4 to 3/4 range your one rod lists is a medium range for most rods and also liable to be the general range most of your baits would fall in (half of what's for sale out there is in this range). But you may want to cast something in the range of 1/8 or even less. The rod might still do it but not as well. Distance would not be as much. It might be harder to get it where you want it because the rod wouldn't load up enough on the backswing as you cast (this helps to launch the bait like a slingshot when it loads up and releases). Also a heavier bait would possibly make the rod feel overloaded as you cast. Some more expensive rods with a lot higher graphite content might even snap under the weight of an overloaded rod tip with too heavy of a bait. It's not as likely to happen as you might think but it can happen, none the less. The lure weight range on the rod is a general guideline to tell you what the rod is best suited for. Not all rods have it though, but most do. They all have a range, just some neglect to list it ( go figure).
That's one reason I like Shimano rods. They list action and power, lure weight and line weight. All of that info is on each rod they make.
I don't mean to confuse you further, but this is probably a good time to let you know that even though your rod says medium "action", it is likely referring to "power" instead. Power is a rod's overall resistance to bending (as examples, a light powered rod is weaker than you have, medium power is in the middle and is what you have, and medium heavy would be stiffer, and so on). Action refers to the point at which the rod tip flexes, such as fast, extra fast, moderate, or slow (none of your rods say this obviously or you would have listed it, I'm sure). You can tell what you have by flexing them some to see, but I'd say your rods are in the fast to maybe medium fast range if I was to guess. Extra fast bends in the top 1/4 length of the rod, fast bends more into the 1/3 range, moderate would bend into the mid section, and slow would be throughout the entire length of the rod blank. Each has it's own use in the fishing world. Fast is generally good for most things. So, consequently, a medium rod with fast tip would be pretty good overall, which is what I'd suspect you have. This is a lot of info to take in, but since we were talking about rod specs, I thought I'd throw it in there. It's not of major importance that you learn it right away, but still good to know about anyway and would give you something to think about for any future rod purchases. Again, not trying to muddy the water, just providing as much info as I can.