Is there a name for the type of shorthand sentence that excludes yourself as the subject? e.g., "Going to the park."

This is an example of pro drop (short for pronoun dropping).

Some languages, like Spanish, pretty much mandate it. Such languages are called pro-drop languages.

In English, most grammarians would probably consider it ungrammatical, but it's pretty standard in informal speech.

Regarding its occurrence in English, Wikipedia writes:

English is considered a non-pro-drop language. Nonetheless, subject pronouns are almost always dropped in imperative sentences (e.g., Come here). In informal speech, pronouns may sometimes be dropped in other types of sentences...

• [Have you] ever been there?
• [I'm] going to the shops. [Do you] want to come?
• Seen on signs: [I am/We are] out to lunch; [I/we will be] back at 1:00 [P.M].

The second and third examples of this passage are precisely the kind of sentences you ask about. These sentences can be called pro-dropped sentences.

Note, however, that pro dropping can occur for any pronominal subject, not just for "I". For example, consider the following dialogue:

A: Where's Ingmar?

B: Went home.

Here, the pronoun "he" is being dropped.

As a final note, these kinds of pro-dropped sentences fall under the more general umbrella of ellipsis:

the omission from speech or writing of a word or words that are superfluous or able to be understood from contextual clues.