Per the wiki article you linked to in your question, secret ballots were introduced to fight things like vote buying, intimidation, and blackmailing.
In principle, one is a public display of opinion (free or not is completely irrelevant), while the other is a private vote -- even if you publicly say that you voted for this or that (freely or not is also irrelevant), no one can check that you actually did vote like that.
On the topic of ballot option endorsements, things are trickier. And I should note in passing that it's not unheard of to sign for a ballot option that you do not approve of, in the interest of giving a voice to the entire political spectrum, and face intimidation before doing so or repercussion after having done so.
In France, for instance, there was a time when the extreme right polled around 10-15% across the whole country, but because they'd rarely have a majority in municipalities and the like, they struggled to get the needed 500 signatures by elected officials to put a Presidential candidate forward. In large part, this was because a lot of intimidation was going on against the mayors who would provide their endorsement in the name of putting all relevant choices on the ballot. The Communist party, by contrast, basically polled epsilon across the country except in a few concentrated areas where they scored very well, and they were on good enough terms with the socialists that they'd get the 500 signatures without any problems.
Countering that problem is a lot trickier.
Frankly the main thing that immediately springs to mind is some kind of first round of voting where only those allowed to endorse candidates would get to vote. But then there's a natural counter argument to do so, which is that: If as a candidate or a promoter of some cause you can't get a large enough number of people to publicly endorse that you're a sensible ballot option, then you or your cause probably do not deserve to be on the ballot at all. (In other words, the exact same type of criteria that guards communities that allow referendums by popular support against voting constantly on pointless topics.)
Another thing that springs to mind is an open ballot. The problem there is that the gate is wide open to interpret whatever voters filled in (think typos, difficult to read handwriting, etc.).