I don't seem to really understand what an attack is unless it directly breaks into or threatens enemy territory, like an invasion, peep, or spike, or threatens to cut/capture. Can anyone give some short hints on the nature of attacking?
You are not alone in this. In fact not understanding this part of the game is a marker for middle kyus and below. You know the way toddlers learn to talk - although some start earlier than others, they all go through the same stages and so you can guess their ages fairly well from just the way they talk. Pros have similar markers for gauging the strength and progress of their pupils.
Since you gave a cluster of words to convey your idea of attacking, I'll suggest a similar way of looking at it but with some significant differences (for example, like jts I disagree with counting invasion as an attack). I don't know your teachers' backgrounds, but I'll assume that, like most western players, they are influenced in some way by the Japanese notion of attacking.
The Japanese for 'to attack' is semeru. There are related words: semai means narrow/confined, semaru means to apply pressure to, and other meanings of semeru (usually written with different characters) include urging someone or torturing someone. In short, there is a feeling of putting pressure on the opponent and keeping that pressure applied by confining him. Crowding him, if you like. Many beginners and even stronger kyu players overlook this and often attack by chasing the enemy out into the open. It may be fun and it may make the opponent respond, but chasing is not semeru - and semeru is better because you keep the pressure on.
So, rather than think about punching, jabbing, poking, kicking or chasing the opponent, think about crowding him. Visualise it perhaps as trying to put a sack over his head. As you will have inferred already, this way of looking at it means you don't always have to be at very close quarters. Go, after all, is the surrounding game.
More specifically, at least when pros use understanding of attacking as a grade marker, it means understanding that the object is not necessarily to capture the opponent. It is sufficient to get some benefit (which may be fairly tenuous, such as getting more thickness).
A Japanese example of semeru is Black 1 below. It aims e.g. at cutting at 'a'.
$$B
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$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . X , X . . . . , . . O X X , . . . |
$$ | . . . . . . . . . . 1 . a O O O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
- Click Here To Show Diagram Code
[go]$$B
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . X , X . . . . , . . O X X , . . . |
$$ | . . . . . . . . . . 1 . a O O O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]
You will see from this example that semeru refers to aiming at an existing weakness. It is not about creating that weakness. Obviously it's good to create weaknesses in the enemy formation, and if you need a word to describe that process you can use 'attack' if it suits you, but it's not normally what is meant by semeru (in Japanese semaru might be used to describe the former process, so there is clearly some kind of relationship, but...).
To make the leap out of the middle or lower kyu bracket you need to learn to appreciate that the effect of Black 1 is not to kill White. The main effect is to get Black 1 as a free move, and with it all the benefits it may bring in future on the
left side of the board. What you also need to appreciate is that it is confining White. Assuming White protects the cutting point at 'a', at some point Black can then harrass the White group further by playing a knight's move on the right side. This is still attacking of the semeru type, but when it gets this close up and nasty it's more often called bullying ijime. Either way, Black is still not expecting to kill the White group. He is just picking up free points.
Now consider starting operations against the White group with that knight's move on the right side (i.e. omitting Black 1). This is not really a semeru attack because it is not confining White. White will certainly respond (somewhere around 1) because he will no doubt feel assailed and threatened, but he is simply being
chased out into the left side which is where Black should be hoping to make profit (because he has invested two unopposed stones there).
So, whatever cluster of words or ideas you choose to associate with attacking in go, you should include the ideas of confining, surrounding and pressure as high priority items, and if you do retain 'chase' in your thesaurus, I'd recommend marking it with an asterisk.