Is it considered fulfilling a promise if only part of it is fulfilled, such as not watching movies except for horror films, or not watching television except for series?

What is considered in a vow is the intention. If you intended to avoid looking at all movies and channels, then you must abandon all of them to fulfill your vow. However, if you intended to avoid only those that are provocative or frivolous, there is no harm in looking at what is beneficial. If there was no intention, then one looks at the reason that led you to make the vow, similar to an oath. If the motive for the vow was to refrain from looking at what is forbidden, then there is no harm in looking at what is free from the forbidden, and this is not considered a breach of the vow; because the context of the statement—which is the motive for the oath—indicates that what is not forbidden was not intended. So, intention affects an oath by specifying and generalizing it, and the reason takes its place when intention is absent. It must be noted that a vow to refrain from looking at what is forbidden is obligatory to fulfill, as it falls under the category of avoiding prohibited acts.