If I SELECT IDs then UPDATE using those IDs, then the UPDATE query is faster than if I would UPDATE using the conditions
I don't think there can be a one straight-forward answer to your "why?" without doing some sort of analysis and research.
The SELECT queries are normally cached, which means that if you run the same SELECT query multiple times, the execution time of the first query is normally greater than the following queries. Please note that this behavior can only be experienced where the SELECT is heavy and not in scenarios where even the first SELECT is much faster. So, in your example it might be that the SELECT took 0.00s because of the caching. The UPDATE queries are using different WHERE clauses and hence it is likely that their execution times are different.
Though the column a is indexed, but it is not necessary that MySQL must be using the index when doing the SELECT or the UPDATE. Please study the EXPLAIN outputs. Also, see the output of SHOW INDEX and check if the "Comment" column reads "disabled" for any indexes? You may read more here - http://dev.mysql.com/doc/refman/5.0/en/show-index.html and http://dev.mysql.com/doc/refman/5.0/en/mysql-indexes.html.
Also, if we ignore the SELECT for a while and focus only on the UPDATE queries, it is obvious that they aren't both using the same WHERE condition - the first one runs on id column and the latter on a. Though both columns are indexed but it does not necessarily mean that all the table indexes perform alike. It is possible that some index is more efficient than the other depending on the size of the index or the datatype of the indexed column or if it is a single- or multiple-column index. There sure might be other reasons but I ain't an expert on it.
Also, I think that the second UPDATE is doing more work in the sense that it might be putting more row-level locks compared to the first UPDATE. It is true that both UPDATES are finally updating the same number of rows. But where in the first update, it is 10 rows that are locked, I think in the second UPDATE, all rows with a as NULL (which is more than 10) are locked before doing the UPDATE. Perhaps MySQL first applies the locking and then runs the LIMIT clause to update only limited records.
Hope the above explanation makes sense!