I’m not sure of the rules. Actually, I’m not even sure if there are any rules. If you have the perseverance, try (but expect you will need to do a visa run anyway).
There is no visa that would allow you to stay for 9 months. There are the following visitor visas: 30-day, 60-day with “no extension” endorsement, 60-day. The last one can be extended twice up to the total of 180 days. So, no more than 6 months then (without a visa run). There is also a resident visa that allows you to apply for an ARC as soon as you enter, and you can then stay as along as the ARC is valid. The visitor visa can be changed to a resident visa while in Taiwan, but it’s not automatic either (this is what you’re trying to do, I understand).
If you enter on a visitor visa, you are allowed to stay for the “duration of stay” period shown on your visa. In your case that means 60 days. Later, when your visa is about to expire (no earlier than 15 days before the expiration date), you can to go to the National Immigration Agency, and ask for extension, which is not automatic. You will need to provide your letter of acceptance, student ID, and current transcript. The allowed duration of your stay will then be extended for another 60 days, unless your semester (quarter) ends earlier.
If the period of your study ends on June 1, and you are already allowed to stay until June 2 or beyond, you will not get any further extension: 1 day to pack your stuff, touch your heart, bye-bye! If you transfer to another institution, you will not get an extension either. You may try though, and it might even work, but have alternative arrangements in case it doesn’t.
Generally, getting an extension is cumbersome, and it always depends on an arbitrary decision of some incompetent random paper pusher. If you have any other options, it’s best avoided.
It’s at the issuing TECO’s discretion. You will only know when you get the passport back: the visa will either say “60 days, no extension” or just “60 days.” Generally, as a tourist you will get a non-extendable visa I think. Maybe unless you invent some really long itinerary and include it with your application (“I want to crawl from Taipei to Pingdong and back”).
If I were you, I’d think like that:
Are you sure you really need the ARC?
Yes: try to get one. It’s still doable. But you might need to leave the country first, get another visa, and prepay for another semester in advance.
No: See point 2.
Do you want to travel around the region (Philippines, Hong Kong, Japan, Malaysia are all waiting), or will you go back to Europe temporarily in the middle of your program anyway?
Yes: use the visa exemption and plan your trips so that you never stay for more than 90 days.
No: think again and if the answer is still no, see point 3.
Get a visitor visa and bravely embrace the bimonthly extension quagmire.