You seem to be asking for yes/no answers to complex questions. Sorry, that is not possible.
Hosting your own server gives you more control over it. In some ways that is more secure. On the other hand, the system administrators you can afford to hire are probably less expert that one ones Siteground can afford, since they host many more sites and can spread the cost.
To demonstrate that Moodle is more secure than another LMS, you would need to do a detailed review of all the code of both systems. (Is the other LMS even open source.) That is probably several person-years of work. So, basically, you can't do that.
Moodle's approach to security is on two levels:
First, general practices that make things naturally secure: the clean_param function for filtering input. Using placeholders to instert values into queries to avoid SQL injection. sesskey to protect against XSRF, ...
Then there are specifics. Lots of bits of the code have been reviewed by different people. Particular security holes are reported to the tracker, and then fixed.
Thank you for your summary of the specifics regarding safeguard practices. This helps me a lot.