What is path and fruition?

In DN2, a king asks the Buddh about the fruits of an ascetic life. The first fruit is described as follows in the context of a former bondworker to the king:

“No, sir. Rather, I would bow to them, rise in their presence, and offer them a seat. I’d invite them to accept robes, alms-food, lodgings, and medicines and supplies for the sick. And I’d arrange for their lawful guarding and protection.”

“What do you think, great king? If this is so, is there a fruit of the ascetic life apparent in the present life or not?”

“Clearly, sir, there is.”

“This is the first fruit of the ascetic life that’s apparent in the present life, which I point out to you.”

In this story, the bondservant takes both parts of your question, as an ordinary person and then as a monk on the path.

Householder Sakyan, interested

Namo tassa bhagavato arahato sammā-sambuddhassa

"What is the attainment of path?", was asked by householder Sakyan.

It's by arising of right view, as a (the first) factor of the path, that the path is attained.

"What is the difference in an ordinary person, a faithful disciple who has not attained any path and the holy one who has attained any of the respective path?", was asked by householder Sakyan.

The ordinary person, a faithful disciple, householder Sakyan, has not gained right view as a (the first) factor of the path, still does not train himself on the Ariya-magga. The Ariya-Puggala, the holy one, has already attained the first factor, right view, as path-element, and trains himself, relaying on right view, in the Noble eightfold path.

It is said, by the Buddha, as transmitted, that a person with firm faith, having gone for refuge, has heard the good Dhamma, follows by faith, is not capable to die without realizing the path. At least when the truth of suffering, aging, sickness and death will be faced, path, by gaining right view, may arise for such a faith-follower.

(Note that this is not given for trade, exchange, stacks and entertainments binding here in this wheel, but as a means to escape from it.)

But what is the attainment of path? How would one describe rightly the attainment of the path?

The path to nibbana has four steps as you mentioned stream entry, once returner, non returner, arhatship. As you have asked How would one describe rightly the attainment of the path? I'll answer to that as below.

What is Stream Entry When someone achieved this state he drops three fetters self-view (sakkāya-ditthi), clinging to rites and rituals (sīlabbata-parāmāsa), and skeptical indecision (Vicikitsa). Ref. But there's a mistake in this. Which is how (sakkāya-ditthi) is defined. It's translated as self-view. With this mistake we are trying to think to attain stream entry we need to drop the myself view. Which is not the correct buddhism.

If someone feels like down voting this, first read this full and question back, before doing that. Because this is the real buddhism

First I'll prove why it cannot be self-view.

1. If someone drops him self in the first stage (drops the fetter of self-view) then who is going to attain the once returner state who is going to follow the rest of the path of Enlightment. To do that there should be a person, there should be a self who is doing that.
2. Dropping fetter mānasaṃ which happens when becoming Arhat cause to drop the self-view not the vise versa. And only after someone becomes Arhat he has no view of him self. He is not going to compare anything with him, as everything is irrelative (opposite of relative) to him (there's no him at this state).

Now let's see what's the real sakkāya means. sakkāya means, Something you are attached (upaadanaya) is worth. As an example, we think nice car is something worth to have, pretty lady / handsome boy is worth, Think about anything that you think it is worth (This can be your children, laptop, parents, pen, pair of shoes, phone with latest Tech etc). Now ask the below questions from your self (If you are not Arhat then there's a self).

1. Why you think that's worth? : Answer would be it exists the way you want. You think the existence of that happens as per your wish. Until that happens, you protect it, you keep it, sometime you archive it saying that had been with you for this much of years (eg: you may keeping a pen which is no longer use, but just because it was given to you by someone special, therefore it's worth for you)
2. Is the same thing worth for some other person as well? : Answer would be No. Why? because it's something you build in your mind. If the real worthiness is in that object (eg: pen, phone, lover, car etc.) it should be worth to everyone. The only answer which explains why everyone doesn't like (doesn't think worth) the same thing, is because its made up in your mind.

So now it should be clear that the real worthiness does not exist in out side world (outside object). Its always something we build in our own mind. Now you should think deeply, how you can expect the same model which you built on your mind will be reflecting in the outside object. (eg: wife thinks, husband will love her forever, man thinks the car will exists as he expects). These are two different things. One in outside and one in your mind. And just because we attached to an outside world (external world) object, then we try to make / control outside world object to make it identical with the model we made it in our mind. But you will not be able to achieve this. This is the real meaning of Anicca not the impermanence.

Now back to sakkāya, It is that we think the external world is worth. And we attached (craving) to that without understanding (not knowing = ignorance) the Anicca behaviour of the world. Therefore to drop the sakkāya you need to understand the Anicca behaviour. Which make you understand nothing is worth in this world, but if you have experienced something is worth (exists according to the same picture built in your mind), it is because that's how it supposed to behave according to the law of nature (which is Pratītyasamutpāda), and not because of the picture in your mind or because of your wish. So when someone understand this he drops the other two fetters which are clinging to rites and rituals (sīlabbata-parāmāsa) and skeptical indecision (Vicikitsa)

What is the difference in an ordinary person, a faithful disciple who has not attained any path and the holy one who has attained any of the respective path?

Answer to the second part of the question: An ordinary person and a faithful disciple who has not attained any path are the same Puthujjana level. And only the person, who has attained the stream entry (Sotāpanna) after knowing the Anicca behaviour of the world, acts differently. Because he knows the truth of the world, to understand that very well and he will not run after attachments as the way he used to be. He has slown down the speed of running after attachments as he knows and understands the truth up to some level.

So stream entry person starts meditating after he become stream entry, and get to the four stages of dhyana as he can deeply concentrate the truth of the world (Anicca) and how world behaves (Pratītyasamutpāda) in dyana state. Which cause him moving to the other three steps and finally attain Nibbana

Hope this clears grey areas in Buddhism.