A lot of good answers, but I haven't yet seen on point addressed: stance.
Some background on stances.
There has been a lot of more-or-less academic* study of how RPGs work and how roleplaying works as an activity. One of the concepts that's been defined is stance, which is roughly defined as your (the player's) relationship to the character you're playing.
What you are describing are two different stances, which is just the fancy way of saying that you have two different possible ways of making decisions. You can decide things in a way that is guided by how you think the character would act based on their experience of events (Actor Stance, so named because you are relating to the character as its actor). You can also decide things in a way that is guided by what directions you'd like to push the unfolding story in, based on your experience as a player of the game-playing events (Author Stance, because you're making choices for the character in order to move them through a story in an interesting way).
There are other stances, but those are the two that you are describing in your question. You can also blend stances, using them to generate ideas for what to do and looking for middle ground.
It's also important to note that "stances" are descriptive. You can try to intentionally take a particular stance, but for the most part it's just a useful way of describing the spectrum of ways roleplayers naturally make decisions for their characters.
*Only amateurly academic. Sadly, I'm unaware of anyone getting paid or accredited in RPG theory.
How stances are relevant to you
Stances aren't incompatible, but which ones you choose (and how you blend them, because they're not exclusive) will determine how you experience the game, what happens, and (through what happens around your character) how you affect the enjoyment of the game for other players. Which one you choose as the guide for decision-making for any given choice (assuming they're in conflict, since when they agree it's not an issue, right?) is really dependent on what sort of game you want to play. What sort of game you want to play also impacts whether you'll enjoy the game you are playing, since they might not match.
Some types of games and their compatible stances
If you are playing "slice of life" games, then Actor Stance is going to be very compatible. In this sort of game, the internal thoughts of the characters, their feelings about events, and the mundane details of their lives are interesting and a significant point of playing the game. Games in which you notice other players consistently leaning toward Actor Stance even when it's results in "bad" decisions by the characters, and nobody is upset that such bad decisions are being made, is likely such a "slice of life" game.
If you are playing an "adventure" game, then Author Stance is going to be primary for some decisions. Mostly, plot-based choices will be made with Author Stance, while characterisation opportunities that don't conflict too much with making good strategic or tactical choices will be guided by Actor Stance or not appear at all. Most games that feature significant combat and "trying to succeed" will be this sort of game. As a rule of thumb, if your characters are part of a team within the game structure (even if not necessarily on a team within the game fiction), then Author Stance is a safe bet for most choices.
(There's a third sort of game that uses another stance I haven't mentioned, and that's where making choices based on things that are entirely outside the characters' experience is okay. An example would be a game where it's okay to say "Oh, the action is over there but my character is stuck at work. I'm… uh, I'm going to say that I need to pick up a package at the post office so I conveniently just happen to walk by the alley where OtherCharacter is being mugged." This is Director Stance and such games feature heavy use of meta-game knowledge to make fun things happen. These exist, but are rarer than games that run on Actor and Author stance, which are ubiquitous.)
Your examples
The senior cop staying in and researching is an example of making the choice of what to do next guided by an Actor Stance. Being skeptical and treating these unusual events as "just another day at work" is what a real person in that situation might do. This kind of play can be a lot of fun, but it also requires a high degree of cooperation: that kind of inward-facing roleplay needs to be given room to happen by the other players, and they have to be the sort of players who find it compelling to being audience to a good bit of roleplaying. Significantly, this will frustrate players and GMs who don't have this kind of slow-paced, mundane story in mind when they set up the game.
The senior cop doing the uncharacteristic thing of running around town poking his nose in things is a more Author Stance sort of choice. You know that this is the beginning of the story, of life being not-as-usual anymore, and engaging with that is entirely valid. In this style of play, characters doing slightly out-of-character things and therefore running into unusual circumstances is why this particular story is being told. Think of it as a novel about an otherwise normal person – their life before and after the book's events is mundane, but the unusual circumstances and choices during the book is why there's a book about these people and choices in the first place. A game that is set up with the assumption that the players will be leaning more to Author Stance tends to have room for Actor Stance, but only if it doesn't prevent the characters from participating in the events of play.
So, your senior cop doing more research could happen in either sort of game. In a more adventure-style game, it might be passed over quickly, with the results (or lack thereof) of the research being quickly handled so that the game can move on. In a more slice-of-life style of game, the GM might linger on mundane descriptions of happenings in the precinct and give you some significant time to just feel out the character with some unstructured roleplaying.
How to apply all this theory to what to do in your games
The key is that the stances can be mixed, but the tolerance for how they're blended will depend on your group and the sort of game you're in. Being aware of the difference is probably the only thing you need to do: play how you will, and if you find that people are getting frustrated with your Author Stance choices, you're probably in a game that expects players to stay within the direct knowledge and mundane sensibilities of your character. If you find that people are getting frustrated with your Actor Stance choices, you're likely in a game that's about unusual choices and adventures where good strategic choices (or at least, attempts at good strategic choices) are important for enjoyable, compatible roleplaying.
If you run into this sort of friction, apologise for the friction and let the rest of the group know that you'll adjust your play to be more strategic or more character-authentic, and to give you more feedback if necessary. Unless they're a pack of jerks who you probably don't want to play with anyway, that level of courtesy should be more than plenty to line up everyone's play styles.
