You are right - it's a continuum. A person at risk applies for asylum and is therefore an asylum seeker. When their application is accepted they are then considered a refugee. They are the same thing but at different points in time.
The "safe routes" concept is largely a fallacy - except for specific circumstances (Hong Kong, Ukraine and latterally some Afghans) it is an impossible ask. Let me explain...
Imagine you are an woman in Afghanistan. I imagine you'll not be feeling very safe or secure right now. You're a doctor. You aren't allowed to work any more, but women keep coming to see you, and you have been warned by the Taliban to desist or else... Now a "safe route" involves you, say, going on line and filling in an application form saying that your life is at risk so you'd like to come to Britain. Around about the time the Taliban intercept your communication, you are dead. Same applies if you "nip to the embassy office". Except of course it closed in 2021 so you need to get to Qatar (which is the UK mission to Afghanistan HQ now). You can't exactly get on a plane!
So you grab your money and belongings and with the help of some male relatives, you travel overland and walk into Pakistan. You are now placed in a huge refugee camp - nearly 1 million Afghans are currently in such camps in Pakistan. According the the UK you are now in a "safe country" and therefore cannot apply for asylum. The only possible way to legally claim asylum is the enter the UK and claim it at the point of entry. You can't afford to get on a plane. You are stuck in Pakistan and are going nowhere. You will never be safe.
There are many versions of stories like that. The only way to enable a safe route would be to allow people to apply from abroad in "safe countries" and provide their reasons for wanting to settle in the UK (often to join up with family for example). The idea that "the small boats" are all young men who are economic migrants is not entirely true. Some will be. But others are genuine asylum seekers fleeing abuse and threats. 72% of "small boat arrivals" between 2018 and 2024 were given refugee status. And refugee status is simply not easy to achieve.
Whether people like it or not, the majority of these people are genuine asylum seekers whose lives are at risk in their home country. The only possible safe route system would need to be allowing people to apply from a safe third country and considering their reasons for wanting to come to the UK in a mature and informed way. That doesn't open the floodgates, as some claim. It starts to put the people traffikkers out of business and allows us to have an informed and humane approach to allowing those at risk in ways that we will (hopefully) never know to settle and build new lives - whether in the UK or elsewhere. It doesn't mean that we have to accept all applications, even where genuine. We still hold the right to refuse entry. But it would allow us to look at supporting those who, for example, already have family here or other good cause to want to settle in the UK instead of them risking their lives further in an attempt to enter by unsafe routes.