Is "fished" a transitive verb in "I fished around in my pocket for my keys"?

In English it doesn't sound natural to say "I fished a fish." You would say "I caught a fish." However, in the instances where I can think of using fish as a verb, it must take a preposition (around, for, in, about).

I went fishing in the lake

good

I went fishing the lake

not good

Does "fish" act as a transitive verb? In English, is there a grammatical term for verbs that require a preposition? Can these verbs still be transitive?

I found this entry in an online dictionary, but honestly I have never heard of the verb fish taking what appears be an object. Let's fish the creek just sounds wrong to my ears.

In some of its senses, fish is a transitive verb, as your dictionary entry attests:

He fished a coin out of his pocket for the boy. ("a coin" is the object)

To count as transitive, a verb must take an object. In your example sentence

I fished [around] [in my pocket] [for my keys].

.. the bracketed constituents are called not objects but "prepositional phrases". "Around" is a complement, or part of the phrasal verb "fished around", according to different grammarians.

"In my pocket" is an adjunct of place, "for my keys" is an adjunct of purpose.

Verbs that do not take direct objects are called "intransitive".

She smiled. (no object)
She smiled at me. (no object; "at me" is a complement)

Many verbs are intransitive in one sense but transitive in another sense, like to fish. So say "I fished a fish" would be ungrammatical because in this particular sense the verb is intransitive.

Actually intransitive is a rather vague term as it can mean used without object (to weep, to laugh, to sleep) or used with a preposition + noun ( to look for sth, to smile at someone). And when a verb can be connected with a preposition + noun it is always the question is it a prepositional object (to the verb, not to the preposition), or a prep + complement or an adverbial sentence part. Such as problem arises eg. with "fishing for compliments". Here I would tend to say "for compliments" is an object, a prepositional object, but I'm sure here opinions are diverging.

By the way "prepositional object" (connected to a verb) is confusing in English because in English terminology any noun after a preposition is called object.