What are the differences between carnitine forms?
Carnitine tartrate is a salt between carnitine (cation) and tartrate (the conjugate base of tartaric acid, anion). So in this case, tartaric acid is not covalently bound to carnitine.

The salt formulation is a little tricky, because carnitine is itself a zwitterion at physiological pH (it has both a positive and negative charge) and so is overall uncharged in pure form. I think the tartrate salt is formed because tartaric acid is more acidic (first pKa = 2.89) than the carboxyl group of carnitine (pKa = 4.09), which leads carnitine to become protonated and therefore +1 charged in a concentrated mixture. The formulation sold by Lonza (which seems to be a major provider) appears to be a ratio of 2 cartinine (68%) to 1 tartrate (32%), which would indicate that tartrate acts as a -2 anion in this mixture (this is a bit surprising as the second pKa of tartaric acid has pKa = 4.40, but maybe I'm missing something).

You can recognize that this is a salt from the naming: the form tartrate indicate a conjugate base of an acid (as in lactate, acetate), and when used in a two-component name like carnitine tartrate (or, sodium acetate) refers to a salt. On the other hand, a carboxylic acid covalently bound to another group is given the suffix -yl, so tartryl carnitine would indicate tartaric acid bound to carnitine, as in fatty acid oxidation.