I'm going to try to do two things in this answer. First, I'm going to try to address the tangle of terms and theories that have got you confused. (So this answer will unfortunately be rather long!) Second, I'll try to address the specific question of 誘う.
Feel free to skip any section that doesn't look immediately helpful :-)
What do '-u verb' and '-ru verb' mean?
The terms '-u verb' and '-ru verb' only make sense in rōmaji:
Verbs such as 泳ぐ can be divided into oyog-u:
The stem is oyog-, which ends in the consonant /g/.
So in linguistics, these are often called 'consonant-stem verbs'.
Verbs such as 食べる can be divided into tabe-ru:
The stem is tabe-, which ends in the vowel /e/.
So in linguistics, these are often called 'vowel-stem verbs'.
-ru is not る
Verbs that end in the kana る aren't always -ru verbs. We saw 食べる tabe-ru above, and that's definitely a -ru verb.
But let's take a look at 図る hakar-u:
As you can see, even though this ends in the kana る, it conjugates like an -u verb.
Unfortunately, this point can be very confusing for people who see the terms -u and -ru without an explanation about rōmaji!
When the verb stem ends in /w/
Your verb, 誘う sasow-, is somewhat special. In Modern Japanese, the consonant /w/ disappears before all vowels except /a/. The conjugation looks like this:
誘います sasow-imasu → saso-imasu (/w/ disappears)
誘う sasow-u → saso-u (/w/ disappears)
誘えば sasow-eba → saso-eba (/w/ disappears)
誘おう sasow-oo → saso-oo (/w/ disappears)
And of course, 誘われる sasow-areru works the same way as 誘わない sasow-anai. The /w/ is followed by /a/, so it doesn't disappear.
一段 and 五段 verbs (the traditional analysis)
In traditional grammar, this was of course analyzed very differently! The smallest unit they divided Japanese into was not the phoneme, but the kana. With kana alone, it wasn't possible for them to indicate that the /w/ of sasow- or the /g/ of oyog- belonged to the stem.
Instead, they naturally concluded that each verb had multiple surface forms to which auxiliaries such as ～ます and ～ない attached.
五段 (Godan) verbs
Let's take a look at your example of 誘う and see how it's traditionally broken down:
Here we find five different kana, わ・い・う・え・お, one from each vowel row. The vowel rows in the kana chart are called 段, so this is traditionally called a 五段 (five-row) verb.
In this analysis, the わ is simply an exception. With other 五段 verbs, you'll find the kana more or less as you expect:
Here we have ま・み・む・め・も, all from the same consonant column. Again we find one kana from each vowel row, so again this is a 五段 (five-row) verb.
一段 (Ichidan) verbs
Next, let's look at how a vowel-stem verb is traditionally broken down:
食べ ＋ない tabe -nai
食べ ＋ます tabe -masu
食べ ＋よう tabe -yoo
Here we just find one kana, べ, which belongs to one vowel column. And so, this is traditionally called an 一段 (one-row) verb.
Of course, there's a lot more to talk about in traditional grammar. My goal here is just to show you where 一段 and 五段 come from. Hopefully this is enough :-)
A terminological mess
Ignoring the theory, regular verbs break down broadly into the following two groups. These groups have been given many names by grammarians, linguists, or just people trying to make things easier for learners:
Verbs like 泳ぐ oyog-u
- 五段 verb
- consonant-stem verb
- -u verb
- Group I verb
Verbs like 食べる tabe-ru
- 一段 verb
- vowel-stem verb
- -ru verb
- Group II verb
But whatever labels and theories you use, they're the same two groups of verbs. I know it's confusing that everyone uses different terms, but I hope I've managed to straighten out all the different terms people throw around!