Actually, in all the examples I've seen of okuri-ashi, neither foot crosses in front of the other.

Generally, in tsugi-ashi, the back foot moves up to just behind the front foot, and then the front foot moves forward. In okuri-ashi, the front foot moves first, and then the back foot is drawn up.

I believe odudog's point was that in your initial description, it sounds like you have the leading foot moving first followed by the following foot. That's not tsugi-ashi, that's okuri-ashi.

If you go to this page, you can see Flash animation examples of okuri-ashi and tsugi-ashi.

Scroll down to 送り足 for okuri-ashi; 継ぎ足 tsugi-ashi is the one below that. The one below that is tsugi-ashi with big strides. Click on 進む to start the animations.