I grade exams together with a colleague but disagree with their grading. What should I do?
Have each person grade different questions, not different students.
Consistency in grading is important, and it is unfair to the students if their grades depend substantially on the allocation of their work to a grader. For this reason, if you must split grading duties with another colleague for a particular assessment item, it is best to split the grading duties for the questions rather than splitting the grading of the students. So, for example, one person grades Q1-3 on all papers and the other person grades Q4-6 on all papers. That way each student is graded by the same person for the same question. (Logistically, each of you should grade your questions on half the papers, then swap.)
It sounds like this ship has already sailed, and you have made the rookie mistake of splitting grading for the students, with different people marking different students. It also sounds like you have tried to discuss this with your colleague, but you have exhausted attempts to change his grading. In that case, even if your own grading style is superior to your colleague, adapting to his grading level for this assessment is probably a reasonable second-best option, simply to maintain consistency of the level of grades awarded. In future, try to avoid the problem all together by splitting grading over questions instead of over students.
It's the instructor of record who is ultimately responsible for the conduct of the course, including grading*. As it sounds like you're not the instructor of record, but rather just someone who is helping grade, it's not really your place to determine what is or is not appropriate grading.
If you have a disagreement with another grader on how to mark exams, and can't resolve it using the information already provided to you (you said the grading template was insufficient to do so), then it's appropriate to take it up with the person in charge of that course (the instructor of record), and see what they say about it.
Now certainly you don't want to bother them about every little grading detail, but if it's a case of large-scale differences, where philosophical differences on how to conduct grading would substantially change students' grades, that's exactly the sort of thing the person in charge of the course should mediate.
Note that things get a little more complex when it's a team-taught course, where there are a number of "primary" instructors. However, in these situations it's normally the case where each instructor takes the lead on a certain topic. As such, they should be considered the primary opinion on issues specific to their topic. (Concerns which cut across multiple topics should be decided by mutual consent of the "primary" instructors.)
*) With the proviso that certain courses have to meet department or accreditation standards. But even in those cases, it's the instructor of record who is responsible for making sure that those standards are followed.
At my institution it is typical to double-mark 20% of submissions for the purposes of standardisation / moderation.
This might mean 80% gets single-marked and 20% gets double-marked, or it might mean that two markers assess 60% each (2x an overlap of 10%). For low-weight assessments this is seen as a recommended best-practice, while being mandatory for high-weight assessments.
If such arrangements for standardising / moderating marks do not exist within your institution, perhaps it is worth suggesting that this (or something similar) is implemented.
Without knowing your department, it is difficult to say whether it is appropriate (or feasible) to suggest this be implemented for your current cohort of submissions, else be part of a push for implementing broader change.
A few notes about the process, for the curious:
Where there is a difference between multiple markers for any individual submission, then it is for the markers to settle upon an agreed mark. Where an agreed mark cannot be settled upon, then it gets escalated within the department for mediation.
Where a significant difference between multiple markers across multiple submissions is identified, then the entire cohort must be double marked (and any differences in mark reconciled as described above). Our benchmark for a "significant difference" is a mean deviation (correction) of >= 7%.