A well-order has a least element in all non-empty subsets, and 1 > 1⁄2 > 1⁄4 > … > 0 has a non-empty subset without a least element, so it’s not a well-order.
In general, every suborder of a well-order is well-ordered. In a word, the property of “being a well-order” is hereditary. (compare: every subset of a finite set is finite)
Your second example, 1 > 1⁄2 > 1⁄4 > … > 0, is a well-order. To make it non-well-ordered, leave out the 0.
A well-order has a least element in all non-empty subsets, and 1 > 1⁄2 > 1⁄4 > … > 0 has a non-empty subset without a least element, so it’s not a well-order.
Yes, you’re right.
Adding to Vladimir_Nesov’s comment:
In general, every suborder of a well-order is well-ordered. In a word, the property of “being a well-order” is hereditary. (compare: every subset of a finite set is finite)