Comments on the suggestions so far:
There were 7 contests in which a 3rd place wasn't awarded. Awarding Gold (1st), Silver (2nd) and Bronze (3rd) ought to be standard in every contest - this alone would ensure that more designers get to see their name in the top 3 slots.
In the case that 2 of the same designer's levels make it into the top 3, either choose only the one that got the most votes or (probably better) count them as a joint award for that slot, and award the next slot to the designer with the next highest amount of votes. Or, to put it another way, we're awarding the designer rather than the individual levels in this particular case, and so we can place equal recognition on both levels. However it ends up being formulated, the end result should be that no fewer than 3 designers end up in the final round.
Making sure that the top three are always awarded should be no problem.
I am also open for puting the focus more on the designer rather than the level for the final placement.
For LOTY specifically I would like to keep the focus on the individual levels though.
If keeping the "3 rule" format, allow all 3 winners of the previous contest to choose 1 rule each, rather than the top spot winner choosing all 3. This in itself would likely incentivise more participation; making it into the top 3 suddenly means more, and a designer who manages to make it that far is rewarded for doing so. This is particularly meaningful when there are only 1 or 2 votes in the difference between the top 3, which is sometimes the case.
Also open for this and I think this is a good thing.
Alternatively, the "3 rule" format could either be reduced to 2 rules (maintain an element of choice whilst reducing workload and possible confusion for participants) or scrapped altogether in favour of a single rule (easiest to follow/maintain and could help make contests shorter, so they could be run more often and therefore provide more chances to win, but could also have the undesired side-effect of reducing participation due to lack of interest in the rule).
I would oppose puting the rule count down to 1 as this can severely reduce participation when the rule is not to everyone's liking.
I tend to say that the rule count is not the main thing to adress your issue.
Also more frequent contests can be counterproductive --> design fatique. So I would be careful here.
In many cases, LDC levels that placed in the top 3 also end up in the LOTY top 3 for that year. Perhaps implement a contest entry condition preventing the same level from entering 2 different contests. The level has already been awarded an accolade previously - giving it another award, whilst perhaps deserving, prevents other designers/levels from receiving recognition. Preventing this is an easy way to ensure that different levels (and, perhaps therefore different designers) get a fresh chance with each contest. And, if the same designer is awarded a LOTY top 3 place for a different level than their LDC top 3 place, it's a chance for that designer to be recognised for more of their work, arguably making the award more valuable to them personally. Conversely, perhaps we deem it necessary to bestow more recognition on a particularly good LDC winner. One way to achieve this would be to have a separate "Contest Level Of The Year" award, which pits all of that year's contest winners against each other for one final vote.
Here I stand strongly by the term "LEVEL of the year". Doesn't matter where that level originates.
I would also not want to split the contest up.
In my opinion, one of the main reasons (besides level design experience) why mostly the same names pop up: Consistency in terms of entering the contests.
In my case, I simply entered every single contest and prepared levels for it. Even if I missed the final round regularly, just the fact I enetred every single time and threw my name+level in the ring makes a huge impact on the result.