Initiative is a concept; however, any roll made for it is a check
Initiative references using checks when determining its value
Typically, you’ll roll a Perception check to determine your initiative [...] some other type of check. [...] Stealth check. A social encounter could call for a Deception or Diplomacy check.
Additionally, the following line shows that Initiative rolls are checks, just with modified rules.
Unlike a typical check, where the result is compared to a DC, the results of initiative rolls are ranked.
Initiative is not a check
The rules on checks describe exactly what a check is (emphasis mine):
- Roll a d20 and identify the modifiers, bonuses, and penalties that apply.
- Calculate the result.
- Compare the result to the difficulty class (DC).
- Determine the degree of success and the effect.
Anything which does not follow this is not a check. Initiative does not have a difficulty class (requirement #3) and does not have degrees of success (requirement #4). Under the definition of a check, initiative is not one.
Additionally, the rules on initiative consistently refer to an "initiative roll", but never an "initiative check". This is contrary to the more general treatment of checks throughout the rules, which are explicitly described as checks.
Initiative is a step in an encounter.
So what is initiative? According to the rules on encounters, initiative is the first step in the structure of an encounter. Although a roll is necessary to complete this step, initiative is not that roll.
The same is true for the other steps. For example, the second step is to "play a round". Although a player on their turn may make several checks, none of those checks is a "play a round check". Even though rolls happen and may be necessary to play a round, playing a round is not a check.
Initiative may involve checks
Rolling for initiative says:
Typically, you’ll roll a Perception check to determine your initiative [...] Sometimes, though, the GM might call on you to roll some other type of check. [...] if you were Avoiding Notice during exploration (page 479), you’d roll a Stealth check. A social encounter could call for a Deception or Diplomacy check.
As a part of the roll initiative step of an encounter, participants may make a check. The rules of that step specify the most common checks (Perception, Stealth, Diplomacy). These are explicitly called out as being checks and as being exceptions to the normal process of a check.