Is it correct to infer a second, third etc place in board games with a score?
Tl;dr: Players' meta-goals do affect gameplay. And unless all players have the same goals, the entire concept of "best player" isn't well defined, let alone "second best", etc.

First a bit of game-theory framework. Let's say each player has a "utility function" which takes as input the game end-state and gives as output how much "utility" they get - an abstract stand-in for happiness, bragging rights, etc. As an example, you might assign yourself 10/1/0 utils for getting 1st/2nd/3rd highest score in the game. A player's ultimate goal is solely to maximize their own expected utility, regardless of how much their opponents get. (Most utility functions in most games, however, are set up so that maximizing your own utility also minimizes your opponents' - e.g. if all players are using that example function, then to maximize yours you have to get more in-game points than everyone else, which minimizes theirs.)

Your choice of utility function absolutely affects gameplay. You give a bunch of accurate examples of this, so I won't go into it.

Now to the philosophy.

There isn't one "correct" utility function. Some games may have been made with one in mind - as you mention, some rules emphasize the winner while others tell you how to rank the players. But if one player is almost as happy with 2nd as 1st and another only really celebrates when they win outright, neither is likely to convince the other that they're wrong. Not only that but in most situations there's no need to - these two can happily play at the same table, each pursuing their slightly different goals. (Though as a sidenote, if these differences in goals remain hidden then some moves may appear irrational. This can lead to hard feelings, so it can be a good idea to discuss where you stand on this spectrum before you play.)

However, players who are playing with different functions are effectively playing (perhaps only slightly) different games. After all, since function affects gameplay, what is objectively a good move for one player might be a bad move for another! Thus if your goal in "inferring positions beyond the winner" is to determine how good the players are at the game relative to each other, well you can't even be certain about inferring that the winner is the best, let alone the other places!

As a concrete example, let's say you have a 3-player game in which Alice is average - say she can get 1st/2nd/3rd place 30%/40%/30% of the time. Bob starts out objectively better - he can get 1st/2nd/3rd place 35%/60%/5%. Then Alice decides she hates getting second and from now on makes a desperation push for 1st every single time, which fails more often than not, so her new values are 40%/0%/60%. I will now tell two stories. In the first story, Bob knows how to do the desperation move but he's happy with second place so he doesn't think it's worth it. Thus Alice wins more often than Bob (40% vs 35%), but Bob is still the better player. In the second story, there's a massive cash prize awarded to first place, and Bob can't figure out how to pull off that desperation move himself - he watches in frustration as Alice walks away with the prize 40% of the time. In this story, the percentages are exactly the same but this time Alice is clearly the better player.

Thus, unless you have a way of forcing everyone to use the same utility function, you might be unable to determine any rank - even best player - since the entire concept of best player isn't well defined. Luckily, this isn't usually a big problem since the people who really want to rank the players often do have a way of forcing everyone to use (roughly) the same utility function: those people are tournament hosts and that way is prizes. So when you're playing for fun you're welcome to your own concept of victory, and when you're in a tournament, well the prize for 1st in the second story above is not only the reward for being the best but also the reason we can define who's the best.