As you hit on, the definition of flexibility is paramount. Rather than debate a proper definition, I'll give some takeaways I use with my clients.
The quick timeframe examples
First, flexibility can improve virtually instantaneously. For instance, in those with pain, it's common for them to be "guarding" a given motion.
Say you have someone who has lower back pain when they lean over. Because of this, when you assess their ability to hip hinge, they only go e.g. 25% into the range of motion.
One might say "they have stiff hamstrings." However, if you change how this person leans over e.g. you don't allow the lower back to round as much, or if you put a chair in front of them so they have some support and lessen the load on the spine, they might suddenly have 50% more range of motion. By changing the motion to be more lower back friendly, or doing something to give the person some peace of mind, range of motion can dramatically improve in real time.
This isn't typically what people think of with flexibility, but it's actually rather common. Point being: in that small time frame nothing muscularly was changed- we didn't acquire a new adaptation that quickly. Just like getting stronger, you don't change sarcomeres in seconds.
But wait! You can change the microscopic instantaneously too. For example, you can tear an ACL functionally without tearing it physically. That is, you can have a traumatic event happen to your knee, causing such a stretch on the ligament, that while it doesn't tear, it's no longer capable of responding to a stretch stimulus. It has no "rebound" left in the material. Called a plastic deformation.
The longer timeframe examples
For the more conventional, where a person has say, a stiff lower back, and they want to know how long it'll take to not feel stiff, I usually tell them 4-6 weeks to notice improvement. Here is why:
Even if you’re actively doing everything better, you're stretching every day, doing enough volume to garner an adaption, passively you aren’t.
What does that mean? To be more clear, we’ll say actively means you consciously manipulating your body (this includes modifying any activities of daily living which may be holding your lower back in a shortened position / causing it to feel stiff), and passively is unconscious. Not only do muscles work actively, they also work passively. They’re always pulling.
The size of a muscle is a great indicator for how much it’s pulling on something i.e. how much tension it’s generating.
Let’s say your lower back muscles are working overtime, but your abdominal muscles undertime. A common example of this is an anterior pelvic tilt. The lower back muscles pull the back of the pelvis up, while certain abdominal muscles pull the front of the pelvis up. They counteract one another.
We could likely say the lower back muscles are excessively hypertrophied and the abdominals not hypertrophied enough. So, even if you’re avoiding an anterior pelvic tilt (or whatever) ALL the time, those lower back muscles are still pulling on your lower back more than you want. It’s simply going to take time for
1) The abdominal muscles to hypertrophy and catch up and or
2) The lower back muscles to atrophy and calm down.
In other words, you can view this similarly to how long does it take to acquire noteworthy hypertrophy: 4-6 weeks.
If you look deeper, you can notice hypertrophy quicker than that, but this a commonly accepted benchmark outside of the microscopic level. Said another way: you can see improvement in strength, flexibility, muscle size, often only a few days apart. For instance, a person comes in for a session one day, a few days later it's certainly plausible they're a bit stronger, more flexible, whatever.
But those aren't usually big jumps. Instead, when someone looks back a month or more, the change is a lot more obvious.
You can flip this to look at atrophy too. If you don't do anything on your legs for a week or two, you can definitely notice atrophy happen that quickly. Astronauts are a good example of this. Two weeks in microgravity and they can be 10% weaker. (Deeper dive on astronauts here.) But you have to look a fair amount harder than at the 4-6 week mark, where it just jumps out at you. (Your average everyday person isn't going to think much of a 10% difference.)
That's a long way of saying the definition of flexibility can be hashed out, but the definition of improvement can too.
Personally, whenever clients ask me how long until they notice improvement, I go with the above for a rationale of 4-6 weeks, even though many will notice improvement quicker than that. Because the final point with all the above is we're assuming the person is perfectly adhering to whatever is necessary to acquire an adaptation. A lofty assumption for many.