Whatever position(s) a player gains during a season, it carries over to the next. In Trumbo's case, he would carry all 3 next year because he met the requirements for eligibility this year. However, if a player is listed at one position at the start of the year and doesn't play there at all that season, he would lose it the following year. For example, Machado is currently listed #$%$ only, but has played only 3B so far. If he only plays 3B this year, he would lose SS eligibility for next year but still have 3B.
Long story short, whatever a player gains, he won't lose the next year. But whatever he starts with, he must play enough, otherwise he loses it the next year.
Mostly everything John said is true, but the staff for Yahoo do have the right to tweak things based on the projected or likely position that a player will contribute to, so if Yahoo decided that it is likely that Machado will win the SS role or play a significant amount of time there, they may (or may not) give him SS eligibility at the beginning of next year, even if he didn't meet the requirements this year. EXAMPLE: Kendrys Morales gaining OF status even though he hadn't met the requirement his recent years.