I have published some papers in computational linguistics and I'm familiar with the literature on tense. First of all, you are right. There are indeed some linguists that do spend time with the distinction between primary tense represented by a word form such as "am" in "I am here" and primary tense represented by an auxiliary as "am" in "I am cooking", and treat these two representations as two different but related linguistic phenomena under the names of "tense" and "aspect". This group of linguists tends to be seen by their peers as "the formalists/structuralists", this distinction being one of many symptoms of caring more about word form and structure than about the meanings that these forms and structures represent.
Other linguists such as Halliday and Matthiessen (Halliday's Introduction to Functional Linguistics, 2014) prefer to organize the contrastive options of language as systems of options. In this sense, for relational processes such as being and having, there are three primary tenses in English: "I was here", "I am here", "I will be here". For action processes such as cooking, the primary present tense has an auxiliary that is not needed for relational processes: "I cooked", "I am cooking", "I will cook". "I cook", on the other hand, represents a habit and, as a consequence, a capability. It represents, therefore, not only a present tense, but also habitualness. "I am going to cook" and "I am about to cook" are other two ways in which the primary future tense is realized in English. There is, however, the possibility of representing a "secondary tense", that is, to represent the time of an action not directly in relation to now, but in relation to another time, that in its turn is relative to now. That is the reason why we can say "I told him, I had cooked", "I told him, I was cooking", and "I told him, I would (was going to/was about to) cook". Linguists that organize occurring wordings in terms of systems of options are known as "systemicists/functionalists" and tend to prefer the terms "primary tense"/"secondary tense" to "tense"/"aspect" in order to avoid confusion with the terminology of formalists/structuralists. But as everything in a society, these are tendencies and trends and never absolutes. You will find for sure formalists/structuralists that use "primary/secondary tense" and functionalists/systemicists that use "tense/aspect". And my short explanation above of what is discussed in linguistics includes only two of many uses of such terms.
This being said, if you want to write about tense in German, I would advise you to state the definition of your terminology either in a formal/structural or in a systemic/functional way. As a tip, if you are making a corpus study, you should pay attention to the fact that it is the past tense in German that tends to be represented by a word form for relational processes such as "war" in "ich war hier" and "hatte" in "ich hatte eine Katze" and by an auxiliary for action processes such as "war" in "ich war einkaufen" and "habe" in "ich habe eingekauft". Another tip is to notice that German has a primary past tense and a primary non-past tense. Future and present tenses are refinements of the non-past tense. In other words, the future primary tenses of "ich werde/gehe kochen" and "ich koche sofort/gleich/bald" are refinements of the non-past primary tense of "ich koche".
However, if you are just learning the language, do not let grammatical terminology get in the way of you learning what the wordings mean. And a good reason for not making a big deal out of grammatical terminology is that it is absolutely irrelevant whether "habe" in "ich habe das getan" is called "perfect tense" or "present tense + perfect aspect" or "finite primary-past auxiliary" when you speak or write in the language. Pay attention to the form/structure only for recognizing and for producing them and pay attention to their meaning for interpreting and formulating clauses. After you make the link between pattern and meaning, you can pick the grammatical theory of your choice for explaining the meaning-making process.