The rules you've learned are OK, but seem a bit restrictive. In fact whenever there's an infinitive and a clitic pronoun, the pronoun goes after the infinitive (regardless of whether there is a conjugated verb before the infinitive, as in rule 2), except in the case of clitic climbing (see below).
So for example, if there's a preposition like a, sin, por, para, en, etc., plus an infinitive, the pronoun always goes after the infinitive:
- Sólo vine a pagarte.
- Salió sin verlo.
- Limpié la casa para venderla.
- Gracias por ayudarnos.
- Mucho gusto en conocerte.
If the subject of the phrase is the infinitive with its arguments (pronouns and complements of time, manner, etc.), then the pronouns also go after the infinitive:
- Es imposible soportarlo en silencio.
- Acercarnos ahora a él será muy difícil.
In the phrase Recuérdame amándote (which is OK though a bit farfetched) the pronoun te doesn't belong with recuerda but with the gerund amando. Gerunds work with pronouns more or less in the same way as infinitives.
The only instance where an infinitive and a pronoun that refers to one of its objects can become separated is illustrated by your rule 2. This happens with some patterns where a main verb (such as querer, poder, intentar) is usually followed by an infinitive. In these cases the pronoun might "climb" from after the subordinated infinitive up to before the main verb. This clitic climbing is an example of something called raising in linguistics.
- Quiero comprarlo. → Lo quiero comprar.
- Intento entenderte. → Te intento entender.
- Podemos ayudarla. → La podemos ayudar.
This pattern is rather restricted. It doesn't work with most of the verbs that are customarily followed by infinitives. The following are all wrong:
- Odiaría verlo. → ❌ Lo odiaría ver.
- Me gustaría tenerlo. → ❌ Me lo gustaría tener.
And conversely, these transformations in the opposite direction are also wrong:
- Lo vi salir. → ❌ Vi salirlo.
- Me escuchó entrar. → ❌ Escuchó entrarme.