I am building a new homepage template following along (sorta) the first tutorial.
Having copied the Homepage.php from the tutorial, and written it to mysite/code I then did http://localurl/db/build?flush=1 with the following results. (I'll attach a copy of the actual messages)

.... messages that looked OK
.....followed by info about changes that made sense in the light of the change I'd made
.....more OK messages, then

Warning: Cannot modify header information - headers already sent by (output started at C:\Apps\xampplite\htdocs\silverstripe\sapphire\core\model\DatabaseAdmin.php:137) in C:\Apps\xampplite\htdocs\silverstripe\sapphire\core\Debug.php on line 151
Error

The website server has not been able to respond to your request.

And that was all..... Removing homepage.php and rebuilding restores the system to working state, but doesn't get me any further in building a new homepage template.

Where have I gone wrong?

I hope this is helpful.
I'll attach the full error response, but the key part is:
FATAL ERROR: Bad class to singleton() - HomePage
At line 85 in C:\Apps\xampplite\htdocs\silverstripe\sapphire\core\Core.php

user_error(Bad class to singleton() - HomePage,256)
line 85 of Core.php

singleton(HomePage)
line 175 of DatabaseAdmin.php

DatabaseAdmin->doBuild()
line 90 of DatabaseAdmin.php

DatabaseAdmin->build(Array)
line 261 of Controller.php

Controller->run(Array)
line 104 of Director.php

Director::direct(/db/build)
line 158 of main.php

Yea the file attachment system has been completely remodelled in the next version of the forum. As for the error it could be because your class is 'HomePage' but that file you attached has the filename 'Homepage' that would cause SS to get lost.