First off, I was about asking this question on German Language & Usage since this is a feature specific to the German language. Possibly, this feature exists in other languages as well but as far as I know not in English. Anyway, since this question is about linguistics I think this is the right place.
From what I know, transitivity, in linguistics, means whether or not a verb is followed by a direct object (called accusative in German). To thank, for example, is a transitive verb.
I thank you.
In contrast, an intransitive verb must not take a direct object. To die is such a verb.
*He died something.
(In addition, respective to comments: To die is defined as intransitive by Merriam-Webster dictionaries but it seems to be ambitransitive, too, because to die a death is valid transitive use of to die.)
Some verbs can be followed by a direct object but this object is not necessary. These verbs are called ambitransitive. As example, consider the verb to eat.
I am eating.
I ate an apple.
And some transitive verbs additionally have a second object, the indirect object (dative in German). The appropriate term is ditransitive.
*I give you.
I'll give you a chance.
I'll explain it to you.
Since the indirect object is the recipient of the direct object the indirect object can't be defined independently from an direct object, at least in English.
*I give to you.
In German, however, you can put a dative object behind some intransitive verbs. Danken (English to thank) is one of those verbs.
Ich danke dir.
In comparison to I thank you, dir is not an accusative in German and consequently not a direct object. I'd claim that in German it is possible to use an indirect object without a direct object.
To cut a long story short: what is the appropriate term for intransitive verbs which take an indirect object or dative, respectively, though?
I don't understand the discussion about the definition of transitivity. In the German sentence Ich danke dir the verb danken is intransitive. I think Dixon's definition is not relevant to this question. Of course, if it were the definition par excellence then danken would be transitive but before posting this question my research gave no inkling that I had to doubt whether danken is intransitive since it has no accusative object and that's the most accepted definition.
(Perhaps some linguistics will disagree. I'm not very familiar with linguistics but there's one rule I follow in general: If even experts don't agree then I stay with the most accepted definition until someone gives a persuasive counterargument.)
That said, sources that handles the German language only, and therefore using the terms accusative and dative object rather than direct and indirect object, mentions that a verb in its transitive form has to be followed by an accusative object.
That said, in the sentence above danken is intransitive. @jlawler confirms this when saying "Transitivity is not a property of verbs per se, but of verbs in a particular sentence -- or perhaps, better, only of sentences" although I think he actually wanted to say the opposite.
So, these discussion might be interesting but are unrelated to my question: what are verbs called that are intransitive and followed by an dative object as danken in Ich danke dir.