What the site is trying to explain is that both German and English have a simple past tense and a perfect tense and that in the not too distant past, they may have been used similarly. In spoken German, the simple past is quickly falling out of use for most regular verbs, and so the similarity between the languages no longer holds and you cannot use English habits as a clue as to how to use these tenses in German.
Some kind of perfect tense exists in almost all the western and central European languages. Even though they may have started off with similar meanings, they have developed quite differently. For instance, it is still going strong along with the simple past in Spanish, but has completely replaced the simple past tense in spoken French in all cases, were it is now called the "past perfect" as opposed to the English "present perfect." Even with English, the perfect is used slightly more in the UK to refer to events in the recent past, than in the US, where Americans would use the simple perfect for the same event.
In American English, the perfect uses a description of a past event to say something about the present that is not otherwise described. It is only compatible with adverbs describing the present (e.g., "now," "today," etc.); whereas the simple past tense is only compatible with adverbs referring to the past (e.g., "then," "an hour ago," etc.,) Adverbial phrases like "just now" or "just" straddle the line between past and present and so might be used with both tenses by some speakers, again probably depending on region.
Since the American perfect uses a past action to describe a present situation, it actually makes no precise claims about what is going on in the present. It only asserts that there is a present reality resulting from the past action. The correct interpretation of what that reality is depends on context and the type of verb.
Often, either tense could be used to describe the same reality, but the speaker and listener would tend to follow up differently, since the two tenses tend to have different implications.
For example, if I were in a class and someone stuck their head in the door and said: "The fire alarm has gone off," I would try to listen for the alarm myself, since that could be the present situation implied by the person using the perfect. However, my school might also have a set protocol to follow as a result of any fire alarm, and so I might think the speaker was simply stating that the protocol was now in effect and needed to be followed. The only certain thing communicated is that the past event has created a present circumstance that is relevant to what I should do.
If the same speaker said, "the fire alarm went off," I would look to the demeanor of the person who said it. If it was said in an urgent tone, I would take this as a call to immediate action. If it was said in a complaining tone, I would look to others first to decide if any immediate action was still relevant. The complaining tone might indicate that the event happened hours ago and might no longer be relevant.
If the same speaker said: "The fire alarm just went off," I would listen for it, since something very recent is being described, but might think that the alarm sounded only briefly and might no longer be audible. I would also judge that no one might have yet had time to judge the urgency of the matter and would probably take the most cautious approach. Time might be of the essence, and delay might be dangerous. It's a recent past event whose implications for the present are uncertain. You have to judge from the circumstances or ask for more information.
Often, either tense can be used for the same pragmatic meaning. If you wake up in the morning and your spouse looks out the window in a northern climate during the winter, they might say: "Oh, it snowed" or "Oh, it has snowed." The first refers to a past event that the spouse presumed happened because of something they see (probably snow on the ground). The second refers specifically to some present condition caused by a past snowfall (again, probably the snow on the ground." I think in the US, most people would say the first, and in the UK most people would say the second in contracted form (Oh, it's snowed).
The difference between the tenses is usually more distinct with stative verbs, where the perfect tends to indicate that the state is still persisting (e.g., the traffic has gotten bad on the roads, leave now or you will be late) and the simple past has no implication at all about the present and often signals that the condition has passed (e.g., the traffic got bad on the roads, but it's clear now and you should have no problems)