Twitter has a bunch of notification methods. The twitter.com website has settings for email notifications and for “web notifications” (browser notifications). This page: How to See Twitter Notifications For Mentions, But Not Likes or Retweets (September 2017) has a bunch of methods that are relevant for Twitter's mobile app (iOS or Android), for the twitter.com website, and for other third-party Twitter apps.
However, none of these apply to https://twitter.com/i/notifications (called the “notifications timeline”) that the question here is about. For example, https://twitter.com/settings/web_notifications says:

and the official help page about the notifications timeline doesn't say anything about such filtering either.
So at this point, one's options, if one wants to avoid notifications for likes and retweets, are:
Don't consume notifications from https://twitter.com/i/notifications at all: use email notifications, or a third-party app, and just ignore (or hide) that tab on twitter.com, or
Use an in-browser method to hide these notifications.
I've been trying (2), and this is what I have so far (will update the answer as I make more progress). Type the following into the javascript console:
function hideBad() {
for (let e of document.querySelectorAll([
'.js-activity-retweet', // "Retweeted your Tweet"
'.js-activity-favorite', // "liked your Tweet"
'.js-activity-favorited_mention', // "liked a reply to you"
'.js-activity-retweeted_mention', // "Retweeted a reply to you"
'.js-activity-favorited_retweet', // "liked your Retweet"
'.js-activity-follow', // "followed you"
'.js-activity-generic' // someone liked something else…
'.js-activity-retweeted_retweet' // "Retweeted your Retweet"
].join(','))) {
e.style.display = 'none';
}
}
hideBad();
window.addEventListener('scroll', hideBad);
This will filter out notifications of those types, until the page is reloaded.
This seems to be working so far, so the next step would be to make it a bookmarklet, or a browser extension, or simply add relevant filters to whatever element-hiding extension you may be using. All of those would work; pick your favourite. For example, if you're using Adblock Plus, you can add the following to “My filter list”:
twitter.com##.js-activity-retweet
twitter.com##.js-activity-favorite
twitter.com##.js-activity-favorited_mention
twitter.com##.js-activity-retweeted_mention
twitter.com##.js-activity-favorited_retweet
twitter.com##.js-activity-follow
twitter.com##.js-activity-generic
twitter.com##.js-activity-retweeted_retweet
The main remaining problem is that the number shown next to the “Notifications” icon won't take these into account. One can probably live with that though. Of course, when twitter.com decides to change its CSS, all of this will break.