Is there a name for the practice of dropping pronouns in written speech?

It's not just pronouns that are getting dropped. It's whole chunks of sentences.

Will respond when received.

lacks not just its subject I, but also the subject it and the auxiliary verb is from when received. This is the written version of Conversational Deletion, a very common practice in speech, discussed here.

The dropping of I (and other subject pronouns) in English is called “diary drop”, after one of the contexts in which it is most common.

It is distinct from pro-drop (mentioned by @BillFranke), in Italian and other “null subject languages”, in that it cannot occur, for instance, in subordinate clauses:

Think (that) I have understood

* Think (that) have understood

Credo che ho capito [word-for-word translation of starred sentence]

There's an article in Wikipedia on "Null-subject" languages. It includes a section on "pro-drop" languages. So one possible answer is "pro-drop". Chinese and Japanese both drop the subject and the subject pronoun.

In Japanese, if the subject is is the speaker, it's usually indicated as the topic -- "Watashi wa" ("wa" is a topic marker) -- of the sentence and the rest of the discourse until the next instance of "wa". Until that next "wa", the listener assumes that the speaker is referring to himself/herself. Subjects take the "ga" marker. Japanese sentences are sometimes translated into English as "As for me (topic), I like fish" = "Watashi was, sakana (fish) ga suki (like) desu (it is)." The clearest translation is simply "I like fish". The structure of Japanese is quite different from English. "Fish" is the subject of the main clause in Japanese, but the object in English.

In Chinese, "I want to eat fish" is usually expressed as "Yào (want) chī (eat) yú (fish)" here in Taiwan. The complete sentence is something like "Wǒ (I) yào chī [xiǎo (small) yú] (fish)". The assumption is that the subject of the sentence is "Wǒ" if it's omitted, unless someone asks, for example, "Is Ms Chen here?" If she's not, then the answer is often "Bùzài" (Not here) instead of "Tā bùzài" (She's not here).

I've never heard the term "pro-drop" (Not a significant fact). There may be some other term as well. A linguistics professor would probably know.