The H-1B is, under the statute, one of the subcategories within the H-1, and is covered by the same rules except to the extent they are explicitly distinguished. The H-1B1, another subcategory, is explicitly not included in the carve-out of the H-1 from the rule generally prohibiting immigration intent for nonimmigrant visas, which, so the H-1B1 is not, while other H-1s including the H-1B are, dual-intent visas.)
H-1 nonimmigrant visas are the set of visas governed by 8 USC Sec. 1101(a)(15)(h)(i), which included the H-1A [now defunct, formerly Sec 1101(a)(15)(H)(i)(a)], and includes the H-1B [Sec 1101(a)(15)(H)(i)(b)], the H-1B1 [Sec 1101(a)(15)(H)(i)(b1)], and the H-1C [Sec 1101(a)(15)(H)(i)(c)].
The relationship is not the same as ID 1030 to ID 103.