Hi,
from those options go for Mailgun!!
your second option is ugly, website look unprofessional and services are most-likely even worse. Check they basic tier SMTP pay 43€/m. They lack all basic security features, no DKIM, no SPF, no DMARC, no RDNS. Deliverability of this SMTP server will be,.. worse than bad,.. And you pay them,… expensive even for free,… If they are able to sell this, they are not viable partner for me,..
Edit: Mailgun is well know a proven, we used it too, it is a good choice.