The form “Ek!” does not show if it includes the speaker or not. If you want to say it more clearly you can use “Eku ni!” or “Eku vi!” And of course you can add ek to the activity itself: “Ekvojaĝu ni!”
Why "get a move on!" instead of "hurry up!" ?
It's an alternate way to say "Let's go", though "ni iru!" is obviously specific to "let's go", while "ek!" can mean not only "let's go!" (for us) but also "off you go!" (for you).
I imagine that this would be useful for starting running races: tri, du, unu, ek!
Can "ek" be used as the verb "eki"? And if so, what would be the difference between "ek!" and "eku!" ?
Yes, "eki" can be a verb.
"Eku!" by itself is second person imperative so I would only understand it as "You, go!", while "Ek!" could mean that but could also mean "Let's go!" for first person plural (we). You could also say that as "Ni eku!", though.
See also the comment by jxetkubo above.
»"Eku!" by itself is second person imperative«
Why do you think that way? I guess because in English you can only say an imperative in third person singular and plural.
“Eku!” is just the imperative form. You can even say “Mi eku!” which can be best translated as “I (really) should go!”.
Sure, you can say that, with an explicit "mi". But if you leave off the subject ("mi, vi, ni, ...") then the implied subject is always "vi" - that's what I meant that the form by itself is considered second person.
This may have to do with that imperative verbs are by default second person in many languages, from Dutch to Japanese.
I understand "Ek!" as "C'est parti !" in French, which WR translates into English as "We're off!", and LEO into German as "Los geht's!". Am I right?
Could this also be used as a general term to shout when someones procrastinating rather than just "go"
I feel so like an army captain would say that to the soldiers when he wants them to go, or even camp instructors to kids while clapping hands! XD Sounds very suitable