Yes, both of these scenarios are quite plausible.
Our moon influences the tides so much because it is very large compared to other natural satellites in our solar system and is fairly close to Earth.
Check out the table on this wikipedia page : https://en.wikipedia.org/wiki/List_of_natural_satellites
Sort them by 'mean radius' and you can see that our moon is the 5th largest in the solar system. It is also closer to Earth (smaller Semi-major axis) than any of the larger ones are to their planet.
There are moons that are much closer to their host planet than our moon is to us, but most of them are significantly smaller.
The total gravitational effect they have on the planet is therefore much less than our moon's, which means way less tidal forces.
In order for the new moon to only come around every few months, it would need a highly eccentric orbit.
Check out the list of the objects on that page, note that Halley's Comet has an eccentricity of ~0.96 compared to the moon's ~0.05.
Check out this page to see that visualized:
So your new moon would have to be quite small, probably only a few 100 kms across at max, and would spend most of its orbit far away from the planet. Then it would come screaming in, pass close by to the planet quickly, and get flung back out again.
As a result, your new moon probably won't be super visible, and it probably won't have a stable long term orbit. It would probably more closely resemble a captured asteroid than an actual moon.