I think there is not much purpose to being overly "strategic" in these circumstances. You certainly don't want to undermine yourself, but let's consider some scenarios:
First, let's say that the professor believes you're a great candidate for the job, someone they'd really like to have in their lab. In this scenario, in their mind you're also a great candidate for other positions. They previously told you they could not offer you a position, so of course they would expect you to pursue other opportunities.
In this scenario, any reasonable asks you have for more time (though, as others point out, you haven't been given any demand for an immediate final response) should be granted. Ultimately, there will have to be some balance (the professor can't wait indefinitely, they will need to look for other candidates if you go elsewhere), but if they respect you, value your best interests, and also truly believe they are making you a good offer, it will be worth a wait for you to make the right decision for you. In this scenario, they'll also understand that many factors go into choosing a PhD position or any job: sometimes these decisions are made based on family or living conditions rather than purely the work (project, supervisor, institution). Far better to make the decision that works for you rather than get in a situation where you are unhappy and leaving early.
In this scenario, you might as well be honest. You don't need to say explicitly "you're not my first choice", and it doesn't even seem like that's the case. You can simply say that you're considering other opportunities, would prefer to make a decision once you have all possible offers to consider, and ask for a deadline for a decision ("I'm applying for other positions and would ideally like to have all the possible options before making my decision; when do you need a definitive answer by?"). Of course, all of this can happen after you've taken Buffy's advice to inquire further about the position.
In the second scenario, there may be some overlap with the first scenario, but there are some caveats. It could be that the professor believes they are the most beautiful and talented gift to the world, and that anyone willing to consider working elsewhere is a fool. Perhaps they're surprised you even applied to other positions when they expected you to sit and wait for them to contact you. Do you want to work for this person who will continue to think this way in all your future interactions with them, including when you eventually finish working with them and apply for a job somewhere else?
Alternatively, they could fear you will go somewhere else and are willing to apply salesmanship tricks like pressuring an immediate decision to manipulate you to choose them before you get another opportunity. Do you want to work for this person, who doesn't actually think they are making a good offer?
I guess in these scenarios it may be worth playing some game if the end goal is to get a position, but I'd recommend thinking about longer term goals like having a good PhD experience that carries you to your next position. In that case, I think the short-term benefits of winning a game are not actually helpful and may be counterproductive.
